Proof of the $K(π,1)$ conjecture for affine Artin groups
Giovanni Paolini Mario Salvetti
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theoremTheorem[section]
definition[theorem]Definition
remark[theorem]Remark
example[theorem]Example
lemma[theorem]Lemma
proposition[theorem]Proposition
corollary[theorem]Corollary
conjecture[theorem]Conjecture
matching[theorem]Matching

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Proof of the K(π, 1) conjecture for affine Artin groups]Proof of the K(π, 1) conjecture
 for affine Artin groups

Giovanni Paolini]Giovanni Paolini
Giovanni Paolini: Department of Mathematics, University of Fribourg, Switzerland (currently at California Institute of Technology, Pasadena, California, USA).




Mario Salvetti]Mario Salvetti
Mario Salvetti: Department of Mathematics, University of Pisa, Italy.

	


paolini@caltech.edu, salvetti@dm.unipi.it

0.01cm We are grateful to Pierre Deligne for his remarks and suggestions on the first version of this paper.
We are also grateful to Emanuele Delucchi and Alessandro Iraci for the useful discussions, and to the anonymous referee for the helpful comments.
A preliminary version of <Ref> is part of the first author's Ph.D. thesis at Scuola Normale Superiore <cit.>, written under the supervision of the second author.
This work was also supported by the Swiss National Science
Foundation Professorship grant PP00P2_179110/1, by Ministero dell'Istruzione, dell'Università e della Ricerca, Prog. PRIN 2017YRA3LK_005, Moduli and Lie Theory
and by the University of Pisa, Prog. PRA_2018_22, Geometria e topologia delle varietà.
0.3cm
The final publication in Inventiones mathematicae is available at <https://doi.org/10.1007/s00222-020-01016-y>.





	We prove the K(π,1) conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space.
	This is a long-standing problem, due to Arnol'd, Pham, and Thom.
	Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions.
	In particular: we show that all affine noncrossing partition posets are EL-shellable;
	we use these posets to construct finite classifying spaces for dual affine Artin groups;
	we introduce new CW models for the orbit configuration spaces associated with arbitrary Coxeter groups; we construct finite classifying spaces for the braided crystallographic groups introduced by McCammond and Sulway.
	





[
    [
    Received: date / Accepted: date
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§ INTRODUCTION



The long-standing K(π,1) conjecture for Artin groups states that the orbit configuration space Y_W associated with a Coxeter group W is always a K(G_W,1) space.
Here G_W is the fundamental group of Y_W and is known as the Artin group associated with W.
The conjecture was proved for spherical Artin groups (i.e. if W is finite) by Deligne <cit.>, after being proved by Fox and Neuwirth in the case A_n <cit.> and by Brieskorn in the cases C_n, D_n, G_2, F_4, and I_2(p) <cit.>.

In this paper, we prove the conjecture for the next important family of Artin groups, namely for all affine Artin groups.
Together with Deligne's result, this covers all the cases where W is a Euclidean reflection group.

[<Ref>]
	The K(π,1) conjecture holds for all affine Artin groups.


The K(π,1) conjecture goes back to the pioneering work of Arnol'd, Brieskorn, Pham, and Thom in the '60s (see <cit.>).
After the fundamental contribution of Deligne, the conjecture was proved for the affine Artin groups of type Ã_n, C̃_n <cit.>, B̃_n <cit.>, and G̃_2 <cit.>.
So our paper completes the list of affine Artin groups with the case D̃_n and with all the remaining exceptional cases.
Unlike the proofs for previously known cases, our approach is essentially “case-free,” although some partial results use the classification of reflection groups.
In particular, it also applies to all previously known affine cases.

Besides Euclidean cases, the conjecture was proved for Artin groups of dimension ≤ 2 and for those of FC type <cit.>.
It was also extended to the configuration spaces of finite complex reflection groups and proved in full generality by Bessis <cit.>.

Our results were made possible by recent advances in the theory by McCammond and Sulway <cit.>, which rely on dual Coxeter and Artin groups and on Garside structures (see <Ref>).
In <cit.>, finite-dimensional classifying spaces for affine Artin groups were constructed, but with an infinite number of cells.
Our proof of the K(π,1) conjecture leads to a significant improvement of their construction: we obtain finite classifying spaces for G_W which are homotopy equivalent to the orbit configuration space Y_W.
In doing that, we derive many new geometric and combinatorial side results for affine Coxeter and Artin groups, which we think are interesting by themselves.


The following are some consequences of the K(π,1) conjecture:
affine Artin groups are torsion-free (this already follows from the construction of McCammond and Sulway <cit.>);
they have a classifying space with a finite number of cells (see <cit.>);
the well studied homology and cohomology of Y_W coincides with the homology and cohomology of the corresponding affine Artin group G_W (see <cit.>);
the natural map between the classifying space of an affine Artin monoid and the classifying space of the corresponding Artin group is a homotopy equivalence (see <cit.>).





 §.§ Outline of the proof and future research directions


For every Coxeter group W and Coxeter element w, there is an associated dual Artin group W_w.
It is known to be naturally isomorphic to the standard Artin group G_W, if W is finite <cit.> or affine <cit.>.
When the noncrossing partition poset [1,w] is a lattice, the dual Artin group is a Garside group, and it admits a standard construction of a classifying space K_W ≃ K(W_w, 1) <cit.>.
The poset [1,w] is indeed a lattice if W is finite <cit.>, but this is not always the case if W is affine <cit.>.

In our proof of the K(π,1) conjecture, one of the key points is to show that K_W is a classifying space for W_w, for every affine Coxeter group W, even when [1,w] is not a lattice (<Ref>).
This can come as a surprise since the standard argument to show that K_W is a classifying space heavily relies on the lattice property.

Then we show that K_W is homotopy equivalent to the orbit configuration space Y_W.
For this, we introduce a new family of CW models X_W' ≃ Y_W, which are subcomplexes of K_W (<Ref>).
Differently from the usual models (such as the Salvetti complex <cit.>), the structure of X_W' depends on the dual Artin relations in W_w rather than on the standard Artin relations in G_W.
Using discrete Morse theory, we prove that K_W deformation retracts onto X_W'.
This completes the proof of the K(π,1) conjecture, and at the same time, it gives a new proof that the dual Artin group W_w is naturally isomorphic to the Artin group G_W (in the affine case).

The outlined program passes through several intermediate geometric, combinatorial, and topological results.
For example, an important step in the proof of the deformation retraction K_W ≃ X_W' is to construct an EL-labeling of the affine noncrossing partition poset [1,w].
This and other contributions are summarized in <Ref> below.

Of course, one can hope to use an analog strategy to solve the K(π,1) conjecture in the general case.
However, in order to carry out such program, it seems that a general geometric theory of dual Coxeter groups is required, as well as a combinatorial theory of noncrossing partition posets associated with arbitrary Coxeter groups (for example: are these posets EL-shellable? how can they fail in being lattices?), and perhaps also new developments in Garside theory (can the lattice condition be relaxed?).
These are interesting and potentially promising directions for future research.





 §.§ Summary of additional contributions












As mentioned above, in this paper we prove several results in addition to the K(π,1) conjecture.
Here we list the ones we consider to be the most important and of independent interest.

In <Ref>, we expand the geometric theory of Coxeter elements in affine Coxeter groups, continuing the work started by McCammond and Sulway <cit.>.
The following is one of our many results. Its analog for finite Coxeter groups was proved by Bessis <cit.>.

A[<Ref>]
	
	Every element u in an affine noncrossing partition poset [1,w] is a Coxeter element of the Coxeter subgroup generated by the subposet [1,u].



The next result sheds some light on the combinatorial structure of affine noncrossing partition posets and is the focus of <Ref>.
Its analog for finite noncrossing partition lattices was proved by Athanasiadis, Brady, and Watt <cit.>.

B[<Ref>]
	All affine noncrossing partition posets are EL-shellable.


We should emphasize that the affine setting differs substantially from the finite setting.
For example:
an affine noncrossing partition poset [1,w] is infinite and not necessarily a lattice;
not all elements of [1,w] are parabolic Coxeter elements;
not all reflections belong to [1,w].
For this reason, the theory requires significant novelties in addition to the well-established results for finite Coxeter groups.

In <Ref> we introduce new CW models X_W' for the orbit configuration space Y_W, by gluing together classifying spaces K_W_T≃ K(G_W_T, 1) of spherical parabolic subgroups.
This is done in full generality, for an arbitrary Coxeter group W.

C[<Ref>]
	For every Coxeter group W, the subcomplexes X_W' ⊆ K_W are naturally homotopy equivalent to the orbit configuration space Y_W.


In <Ref> we show that K_W is a classifying space, even when [1,w] is not a lattice.
Our proof makes use of the construction of braided crystallographic groups by McCammond and Sulway <cit.>, a “Garside completion” of dual affine Artin groups.

D[<Ref>]
	For every affine Coxeter group W and Coxeter element w ∈ W, the complex K_W is a classifying space for the dual Artin group W_w.



In <Ref> we show that K_W deformation retracts onto a subcomplex with a finite number of cells.
Without additional effort, this argument also applies to the classifying space of a braided crystallographic group.
We obtain the following consequence.

E[<Ref>]
	Every braided crystallographic group has a classifying space with a finite number of cells.


It is possible that the techniques of <Ref> can be adjusted to braided crystallographic groups, to obtain a smaller classifying space with some interesting geometric interpretation (and perhaps prove a crystallographic version of the K(π,1) conjecture).
This might be part of some bigger picture, where every (dual) Artin group has a crystallographic Garside completion, and their classifying spaces are geometrically related.




 §.§ Structure of this paper


In <Ref> we recall the most important background definitions and results that are needed in the rest of the paper.
In <Ref> we prove several geometric results about Coxeter elements in affine Coxeter groups, expanding the theory of <cit.>.
This section goes hand in hand with Appendix <ref>, where we carry out explicit computations for the four infinite families of irreducible affine Coxeter groups.
<Ref> are mostly independent from each other.
They cover separate intermediate steps of our proof of the K(π,1) conjecture, as described earlier.
Finally, in <Ref>, everything is put together to prove the K(π,1) conjecture.















§ BACKGROUND





 §.§ Coxeter groups and Artin groups




Let W be a Coxeter group, i.e. a group with a presentation of the following form:

    W =  S | (st)^m(s,t) = 1   ∀  s,t ∈ S  such that m(s,t) ≠∞,

where S is a finite set, m(s,s) = 1 for all s ∈ S, and m(s,t) = m(t,s) ∈{2,3,4,…}∪{∞} for all s ≠ t in S.
This presentation can be encoded into a Coxeter graph: the vertices are indexed by S, and there is an edge connecting s and t whenever m(s,t) ≥ 3; this edge is labeled by m(s,t) whenever m(s,t) ≥ 4.
A Coxeter group is irreducible if its Coxeter graph is connected.
Any conjugate of an element of S is called a reflection.
Denote by R ⊆ W the set of reflections.
Any conjugate of the set S is called a set of simple reflections of W, and can be used in place of S to give a presentation of W of the same form as (<ref>), with an isomorphic Coxeter graph.


If S ⊆ R is any set of simple reflections of W (not necessarily the one used to define W), the product of the elements of S in any order is called a Coxeter element of W.
Also, for any subset T ⊆ S, the subgroup W_T generated by T is a parabolic subgroup of W (it is a Coxeter group, and T is a set of simple reflections of W_T).
A Coxeter element of a parabolic subgroup of W is called a parabolic Coxeter element of W.
When a set of simple reflections S is fixed, the parabolic subgroups W_T with T ⊆ S are called standard parabolic subgroups.
The rank (or dimension) of W is the largest cardinality of a subset T ⊆ S such that the parabolic subgroup W_T is finite.
We refer to <cit.> for more background information on Coxeter groups.

We are mostly interested in the case where W is a finite or affine Coxeter group, or equivalently, a (finite or affine) real reflection group.
In this case, W acts faithfully by Euclidean isometries on some affine space E = ^n, and the elements of R act as orthogonal reflections with respect to some hyperplanes of E.
These hyperplanes form a locally finite hyperplane arrangement in E, which we denote by .
The connected components of the complement of  in E are called chambers.
Given a chamber C, its walls are the hyperplanes H ∈ such that H ∩C̅ is a (n-1)-dimensional polyhedron.
The collection of all the chambers forms the Coxeter complex of W.
The smallest possible dimension n which is needed to construct such a representation is equal to the rank of W.
If n is equal to the rank of W, the resulting representation is essential.
If W is an irreducible finite Coxeter group, then all its elements must fix a point of E (so W may as well be realized as a group of linear isometries), and all chambers in an essential representation are unbounded simplicial cones.
If W is an irreducible affine Coxeter group, then all chambers in an essential representation are bounded simplices.
We refer to <cit.> for the definition of root systems, positive systems, simple systems, crystallographic systems, and crystallographic Coxeter groups.

Irreducible affine Coxeter groups are classified into four infinite families (Ã_n, B̃_n, C̃_n, D̃_n) and five exceptional cases (Ẽ_6, Ẽ_7, Ẽ_8, F̃_4, G̃_2).
Their Coxeter graphs are shown in <Ref>.
These groups can all be constructed from the corresponding irreducible crystallographic root systems, as follows (see <cit.>).
For each α in a crystallographic root system Φ⊆^n, and for each integer k ∈, consider the affine hyperplane H_α, k = { x ∈^n | x, α = k }.
Then take the group generated by the orthogonal reflections with respect to the hyperplanes H_α,k.
The corresponding reflection arrangement  is the collection of all these hyperplanes H_α,k.




If W is a finite or affine Coxeter group, acting on E = ^n, define the configuration space Y associated with W as

    Y = ^n ∖⋃_H ∈ H ⊗_.

In other words, this is the complement of the complexification of the hyperplane arrangement .
Then W naturally acts on Y, and its quotient Y_W = Y / W is the orbit configuration space associated with W.
Up to homotopy equivalence, Y and Y_W do not depend on the chosen representation of W as a subgroup of the isometry group (E).
The construction of Y and Y_W can be extended to arbitrary Coxeter groups by considering the Tits cone, see <cit.>.

If S is a set of simple reflections of W, the Artin group associated with the Coxeter group W is

    G_W =  S |stst…_m(s,t)  terms = tsts…_m(s,t)  terms∀  s,t ∈ S  such that m(s,t) ≠∞,

see <cit.>.
It is isomorphic to the fundamental group of the orbit configuration space Y_W <cit.>.


	The orbit configuration space Y_W is a classifying space for the Artin group G_W.


Artin groups associated with finite (resp. affine) Coxeter groups are called spherical (resp. affine).
Prior to this work, the K(π,1) conjecture was proved for spherical Artin groups <cit.>, for affine Artin groups of type Ã_n, C̃_n <cit.>, and B̃_n <cit.>, for Artin groups of dimension ≤ 2 (this includes the affine Artin group of type G̃_2) and of FC type <cit.>.

As shown in <cit.>, the orbit configuration space Y_W has the homotopy type of a CW complex X_W, with one |T|-cell c_T for every T in

    Δ_W = { T ⊆ S |the standard parabolic subgroup W_T is finite}.

In particular, the 1-cells of X_W are indexed by S, and the 2-cells are indexed by the unordered pairs {s,t}⊆ S with m(s,t) ≠∞.
The 1-cells are oriented in such a way that c_{s} corresponds to the generator s of the fundamental group G_W, and a 2-cell c_{s,t} corresponds to the relation stst… = tsts….
In the literature, the CW complex X_W is usually called the Salvetti complex of W.
For T ⊆ S, there is a natural inclusion of complexes X_W_T⊆ X_W, induced by the inclusion Δ_W_T⊆Δ_W.






 §.§ Posets



We now recall some basic terminology about partially ordered sets (posets).
See <cit.> for a more detailed exposition.

Let (P, ≤) be a poset.
If p < q in P and there is no element r∈ P with p<r<q, then we say that q covers p, and write p ⋖ q.
Given an element q ∈ P, define P_≤ q= {p ∈ P | p ≤ q}.
We say that P is bounded if it contains a unique minimal element and a unique maximal element.
A (finite) chain in P is a totally ordered sequence p_0 < p_1 < … < p_n of elements of P.
A chain of n+1 elements is conventionally said to have length n.

If p ≤ q, the interval [p,q] in P is the set of all elements r ∈ P such that p ≤ r ≤ q.
We say that P is graded if, for every p ≤ q, all the maximal chains in [p,q] have the same (finite) length.
Then there exists a rank function P → such that (q) - (p) is the length of any maximal chain in [p,q].
The rank of P is defined as the maximal length of a chain of P.

A poset P is said to be a lattice if every pair of elements p,q ∈ P has a unique maximal common lower bound and a unique minimal common upper bound.

The Hasse diagram of a poset P is the graph with vertex set P and having an edge (p, q) for every covering relation p ⋖ q.
We indicate by (P) = { (p,q) ∈ P × P | p ⋖ q} the set of edges of the Hasse diagram of P.






 §.§ Lexicographic shellability



In this section, we recall the definition of EL-shellability
<cit.>.

Let P be a bounded poset.
An edge labeling of P is a map λ(P) →Λ, where Λ is some poset.
Given an edge labeling λ, each maximal chain c = (x ⋖ z_1 ⋖…⋖ z_t ⋖ y) between any two elements x ≤ y has an associated word

    λ(c) = λ(x, z_1)  λ(z_1, z_2)  … λ(z_t, y).

We say that the chain c is increasing if the associated word λ(c) is strictly increasing.
Maximal chains in a fixed interval [x,y]⊆ P can be compared lexicographically (i.e. by using the lexicographic ordering on the corresponding words).


	Let P be a bounded poset.
	An edge-lexicographic labeling (or simply EL-labeling) of P is an edge labeling such that in each closed interval [x,y] ⊆ P there is a unique increasing maximal chain, and this chain lexicographically precedes all other maximal chains of [x,y].
	



A bounded poset that admits an EL-labeling is said to be EL-shellable.
If P is an EL-shellable poset, then the order complex of P ∖{min(P), max(P) } is homotopy equivalent to a wedge of spheres.

Let P_1 and P_2 be bounded posets that admit EL-labelings λ_1(P_1)→Λ_1 and λ_2(P_2) →Λ_2, respectively. Assume that Λ_1 and Λ_2 are disjoint and totally ordered.
Let λ(P_1 × P_2) →Λ_1 ∪Λ_2 be the edge labeling of P_1× P_2 defined as follows:
lCl
	λ((x,y),   (z,y))    =    λ_1(x,z) 

	λ((x,y),   (x,t))    =    λ_2(y,t).



	
	Fix any shuffle of the total orderings on Λ_1 and Λ_2, to get a total ordering on Λ_1 ∪Λ_2.
	Then the product edge labeling λ defined above is an EL-labeling of P_1 × P_2.








 §.§ Discrete Morse theory



In this section we recall Forman's discrete Morse theory <cit.>.
We follow the point of view of Chari <cit.>, using acyclic matchings instead of discrete Morse functions, and we make use of the generality of <cit.> for the case of infinite CW complexes.

Let P be a graded poset, and denote by H the Hasse diagram of P.
Given a subset  of (P), we can orient all edges of H in the following way: an edge (p,q) ∈(P) (i.e. with p ⋖ q) is oriented from p to q if it is in , otherwise in the opposite direction.
Denote this oriented graph by H_.

A matching on P is a subset ⊆(P) such that every element of P appears in at most one edge of .
A matching  is acyclic if the graph H_ has no directed cycles.
Given a matching  on P, an alternating path is a directed path in H_ such that two consecutive edges of the path do not belong both to (P) ∖.
In the graph H_, the edges in  increase the rank by 1, and the edges in (P) ∖ decrease the rank by 1.
Therefore, a matching  is acyclic if and only if it has no closed alternating paths (which are called alternating cycles).
The elements of P that do not appear in any edge of  are called critical (with respect to the matching ).
An acyclic matching  is proper if, for every p ∈ P, the set of vertices of H_ reachable from p (with a directed path) is finite.


Let X be a CW complex.
The face poset (X) of X is the set of its (open) cells together with the partial order defined by τ≤σ if τ̅⊆σ̅.
For all CW complexes X considered in this paper, the face poset (X) is a graded poset with rank function (σ) = (σ).
Recall that each cell of X has a characteristic map Φ D^n→ X, where D^n = {x ∈^n |x≤ 1 }.

Let σ and τ be cells of X.
If τ⋖σ we say that τ is a face of σ.
We say that τ is a regular face of σ if, in addition, the following two conditions hold (set n=(τ) and let Φ be the characteristic map of σ):

	0.1cm
	
  * Φ|_Φ^-1(τ)Φ^-1(τ) →τ is a homeomorphism;
	
  * Φ^-1(τ) is a closed n-ball in D^n+1.


The following is a particular case of the main theorem of discrete Morse theory and follows from <cit.>.


	
	Let X be a CW complex, and let Y ⊆ X be a subcomplex.
	Suppose that there exists a proper acyclic matching  on the face poset (X) such that: (Y) is the set of critical cells; for every (τ, σ) ∈, we have that τ is a regular face of σ.
	Then X deformation retracts onto Y.
	In particular, the inclusion Y ↪ X is a homotopy equivalence.


We conclude by recalling a standard tool to construct acyclic matchings.


	Let η P → Q be a poset map.
	For all q ∈ Q, assume to have an acyclic matching _q ⊆(P) that involves only elements of the fiber η^-1(q) ⊆ P.
	Then the union of these matchings is itself an acyclic matching on P.





 §.§ Interval groups and Garside structures



We now recall the construction of interval groups, and how they give rise to Garside structures.
Our exposition mostly follows <cit.> and <cit.>.
See <cit.> for a complete reference on Garside structures.

Let G be a group, with a (possibly infinite) generating set R ⊆ G such that R = R^-1.
Suppose that the elements of R are assigned positive weights bounded away from 0 that form a discrete subset of the positive real numbers.
Assume that r and r^-1 have the same weight, for every r ∈ R.
For every x ∈ G, denote by l(x) the minimum sum of the weights of some generators r_1,r_2, …, r_k ∈ R such that r_1r_2… r_k = x.
In other words, l(x) is the distance between 1 and x in the weighted right Cayley graph of G (with respect to the weighted generating set R).

The group G becomes a poset if we set x ≤ y whenever l(x) + l(x^-1y) = l(y), i.e. if there is a minimal length factorization of y that starts with a minimal length factorization of x.
Given an element g ∈ G, denote by [1,g]^G ⊆ G the interval between 1 and g (with respect to the partial order ≤ in G).
The Hasse diagram of [1,g]^G embeds into the Cayley graph of G.
Every edge of the Hasse diagram is of the form (x, xr) for some r ∈ R, and we label it by r.


	The interval group G_g is the group presented as follows.
	Let R_0 be the subset of R consisting of the labels of edges in [1,g]^G.
	The group G_g has R_0 as its generating set, and relations given by all the closed loops inside the Hasse diagram of [1,g]^G.
	


The interval [1,g]^G is balanced if the following condition is satisfied: for every x ∈ G, we have l(x) + l(x^-1g) = l(g) if and only if l(gx^-1) + l(x) = l(g).
This condition is automatically satisfied if the generating set R is closed under conjugation and the weight of a generator is equal to the weight of all its conjugates.


	If the interval [1,g]^G is a balanced lattice, then the group G_g is a Garside group.
	


See <cit.> for the definition of Garside groups.
As in <cit.>, we use the term “Garside group” in the sense of Digne <cit.> (so that the generating set R need not be finite).

The classifying space of a Garside group can be constructed explicitly, as shown in <cit.>.
Here we generalize this construction to the case of arbitrary interval groups arising from balanced intervals, without the lattice assumption.
In the case of Garside groups, it is equivalent to the definitions given in <cit.> (see in particular <cit.>) and in <cit.>.



	Realize the standard d-simplex Δ^d as the set of points (a_1,a_2,…,a_d) ∈^d such that 1 ≥ a_1 ≥ a_2 ≥…≥ a_d ≥ 0.
	The interval complex associated with a balanced interval [1,g]^G is a Δ-complex (in the sense of <cit.>) having a d-simplex [x_1|x_2|…|x_d] for every x_1,x_2,…,x_d ∈ [1,g]^G such that:
	
		
  * x_i ≠ 1 for all i;
		
  * x_1x_2… x_d ∈ [1,g]^G;
		
  * l(x_1x_2… x_d) = l(x_1) + l(x_2) + … + l(x_d).
	
	The faces of a simplex [x_1|…|x_d] are as follows.
	
		
  * The face {1=a_1≥ a_2≥…≥ a_d ≥ 0 } of [x_1|…|x_d] is glued to the (d-1)-simplex [x_2|…| x_d] by sending (1,a_2,…,a_d) to (a_2,…,a_d) ∈Δ^d-1.

		
  * For 1 ≤ i ≤ d-1, the face {1 ≥ a_1 ≥…≥ a_i = a_i+1≥…≥ a_d ≥ 0} of [x_1|…| x_d] is glued to the (d-1)-simplex [x_1|…|x_ix_i+1|…|x_d] by sending (a_1,…,a_i,a_i,a_i+2,…,a_d) to (a_1,…,a_i,a_i+2,…,a_d) ∈Δ^d-1.
		
		
  * Finally, the face {1≥ a_1 ≥…≥ a_d = 0} of [x_1|…|x_d] is glued to the (d-1)-simplex [x_1|…|x_d-1] by sending (a_1,…,a_d-1,0) to (a_1,…,a_d-1) ∈Δ^d-1.
	
	Notice that there is a unique vertex, which is indicated by [ ].
	The 1-simplices are oriented going from 0 to 1 in Δ^1 = [0,1].
	See <Ref> for an example.
	The fact that [1,g]^G is balanced ensures that the faces of a simplex also belong to the interval complex.
	




The fundamental group of the interval complex associated with [1,g]^G is G_g.
This can be easily checked by looking at the 2-skeleton.



	
	If [1,g]^G is a balanced lattice, then G_g is a Garside group and the interval complex associated with [1,g]^G is a classifying space for G_g.









 §.§ Intervals in the group of Euclidean isometries



In this section, we recall the main result of <cit.>.
Let V≅^n be a n-dimensional Euclidean vector space, and let E be the associated affine space (where the origin has been forgotten).
Given an affine subspace B ⊆ E, denote by (B) ⊆ V the space of directions of B.
Given a subset U ⊆ V, denote by (U) the linear subspace generated by U.

Let L = (E) be the group of Euclidean isometries of E.
For every isometry u ∈ L, define its move-set (u) = { u() - |∈ E }⊆ V.
This is an affine subspace of V, and it has a unique vector μ of minimal norm.
Define the min-set of u as (u) = {∈ E | u() =  + μ}⊆ E. This is an affine subspace of E.

An isometry u ∈ L is called elliptic if it fixes at least one point, and hyperbolic otherwise.
If u is elliptic, then (u) is a linear subspace, μ = 0, and (u) coincides with the set of fixed points of u, which we denote by (u).
For every isometry u ∈ L, there is an orthogonal decomposition V = ((u)) ⊕((u)) <cit.>.

The group L is generated by the set R of all orthogonal reflections (where every reflection is assigned a weight of 1).
The length l(u) computed using the generating set R is called the reflection length of u.
If u is elliptic, then l(u) = (u); if u is hyperbolic, then l(u) = (u)+2 <cit.>.


	Define the global poset (P, ≤) as the set containing an element e^B for every affine subspace B ⊆ E, and an element h^M for every non-linear affine subspace M ⊆ V.
	The order relations in P are as follows: h^M ≤ h^M' if M ⊆ M'; e^B ≤ e^B' if B ⊇ B'; e^B < h^M if (M)^⊥⊆(B).
	Define also an invariant map L → P that sends u to e^(u) if u is elliptic, and to h^(u) if u is hyperbolic.
	



	For every isometry u ∈ L, the restriction of the invariant map is a poset isomorphism between the interval [1,u]^L and the model poset P(u) = P_≤ (u).
	






 §.§ Dual Artin groups




In this section we recall the definition and some properties of dual Artin groups associated with a Coxeter group W, focusing on the cases where W is finite or affine.
See <cit.> for the finite case, and <cit.> for the affine case.

Let W be a Coxeter group, R its set of reflections, and S ⊆ R a set of simple reflections.
Assign a weight of 1 to every reflection r ∈ R.
Let w be a Coxeter element, obtained as a product of the elements of S.
The dual Artin group with respect to w is the interval group W_w constructed using R as the generating set of W.

The properties of a dual Artin group are strictly related to the combinatorics of its defining interval [1,w]^W, which in turn depends on the geometry of the Coxeter element w.
The interval [1,w]^W is a graded poset of rank |S|.
As explained in <Ref>, the edges of the Hasse diagram of [1,w]^W are naturally labeled by a subset R_0 ⊆ R.
Maximal chains in [1,w]^W correspond to minimal length factorizations of w as a product of reflections.


Since the set R of reflections is closed under conjugation, it is possible to rewrite factorizations as follows (this is a consequence of the so called Hurwitz action).

	
	Let u = r_1r_2… r_m be a reflection factorization in a Coxeter group W.
	For any selection 1 ≤ i_1 < i_2 < … < i_j ≤ m of positions, there is a length m reflection factorization of u whose first j reflections are r_i_1r_i_2… r_i_j, and another length m reflection factorization of u where these are the last j reflections.


If the Coxeter graph of W is a tree, then all its Coxeter elements are geometrically equivalent,
and give rise to isomorphic intervals [1,w]^W <cit.>.
This is the case for all irreducible finite and affine Coxeter groups except Ã_n.
In the case Ã_n, there are ⌊n+1/2⌋ equivalence classes of Coxeter elements: a choice of representatives is given by (p,q)-bigon Coxeter elements, where (p,q) is a pair or positive integers such that p ≥ q and p+q = n+1 <cit.>.


The generating set R_0 ⊆ R of a dual Artin group contains S (for a general Coxeter group, this follows from <Ref>).
Then there is a natural group homomorphism from the usual Artin group G_W to the dual Artin group W_w.


	
	If W is a finite or affine Coxeter group, the natural homomorphism G_W → W_w is an isomorphism.


It is not known in general whether a dual Artin group is isomorphic to the corresponding Artin group, or whether the isomorphism type of a dual Artin group depends on the chosen Coxeter element w.

One important motivation to introduce dual Artin groups is that sometimes they are Garside groups.
For example, this happens if W is finite.


	If W is a finite Coxeter group, the interval [1,w]^W is a lattice for every Coxeter element w.
	Therefore the dual Artin group W_w is a Garside group.
	


The intervals [1,w]^W that arise from finite Coxeter groups W are well-studied, and are called (generalized) noncrossing partition lattices (see <cit.>).
They are known to be EL-shellable <cit.>.
By analogy with the finite case, for any Coxeter group W and Coxeter element w, we call the interval [1,w]^W a noncrossing partition poset associated with W.


Suppose now that W is an irreducible affine Coxeter group, acting as a reflection group on E = ^n, where n is the rank of W.
The Coxeter element w is a hyperbolic isometry of reflection length n+1, and its min-set is a line ℓ called the Coxeter axis <cit.>.
We gain some insight on the structure of the interval [1,w]^W by comparing it with the interval [1,w]^L in the group of all Euclidean isometries of E (see <Ref>).



	
	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	Then the inclusion [1,w]^W ↪ [1,w]^L is order-preserving and rank-preserving.
	In particular, u≤ v in [1,w]^W implies (u) ≤(v) in the model poset P(w).



	The length functions of W and L agree on the Coxeter element w, so they agree on the entire interval [1,w]^W.
	As a consequence, [1,w]^W ⊆ [1,w]^L.
	In addition, if we have u ≤ v in [1,w]^W, then u^-1v ∈ [1,w]^W and l(u^-1v) = l(v)-l(u).
	Since the length functions agree, we have u ≤ v also in [1,w]^L.
	The last part of the statement follows from <Ref>.


Notice that [1,w]^W is not a subposet of [1,w]^L in general: it is possible to have u, v ∈ [1,w]^W with u ≰v in [1,w]^W, but u ≤ v in [1,w]^L (see <Ref>).
However, if W' is a finite Coxeter group (acting as a reflection group on the vector space V = ^n) and w' is one of its Coxeter elements, then [1,w']^W' is known to be a subposet of [1,w']^(V) <cit.>.


Then, in the affine case, we can show that the condition (u) ≤(v) implies u ≤ v in [1,w]^W whenever u and v are elliptic.
In this case, the condition (u) ≤(v) means (u) ⊇(v).


	
	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	If u,v ∈ [1,w]^W are elliptic elements with (u) ⊇(v), then u ≤ v in [1,w]^W.



	We proceed by induction on l(u), the case l(u)=0 being trivial.
	Suppose from now on that l(u) ≥ 1.
	Let W' be the subgroup of W generated by the reflections in [1,u]^W ∪ [1,v]^W.
	By <cit.> this is a Coxeter group, and its set of reflections contains R ∩ ([1,u]^W ∪ [1,v]^W).
	Every reflection in a minimal length factorization of u (resp. v) in W belongs to [1,u]^W (resp. [1,v]^W), and so it belongs to W'.
	This means that the minimal length factorizations of u and v are the same in W and W'.
	
	For every reflection r in [1,u]^W ∪ [1,v]^W, we have (r) ⊇(v) by <Ref>.
	Therefore every element of W' fixes (v), and so W' is finite.
	By <cit.> (see also <cit.>), every reflection r ∈ W' is part of a minimal length factorization of v in W', and so also in W, thus r ∈ [1,v]^W.
	This proves that R ∩ [1,u]^W ⊆ R ∩ [1,v]^W.
	
	Let r be any reflection in [1,u]^W (there is at least one reflection because l(u) ≥ 1).
	We have (r) ⊇(u) ⊇(v) by <Ref>, and therefore r ≤ u ≤ v in [1,w]^L by <Ref>.
	If we write u = ru' and v = rv', we have u' ≤ v' in [1,w]^L, and so (u') ⊇(v').
	Since r ∈ [1,u]^W, we have u' ∈ [1,u]^W ⊆ [1,w]^W and l(u') = l(u)-1.
	In addition, since r ∈ [1,v]^W, we have v' ∈ [1,v]^W ⊆ [1,w]^W.
	By induction, u' ≤ v' in [1,w]^W.
	This implies that u ≤ v in [1,w]^W.


The direction of the Coxeter axis ℓ is declared to be vertical, and the orthogonal directions are horizontal.
An elliptic isometry is horizontal if it moves every point in a horizontal direction, and it is vertical otherwise <cit.>.
Given u ∈ [1,w]^W, the right complement of u is the unique v∈ [1,w]^W such that uv=w. Define the left complement similarly.

<Ref> and some geometric considerations in <cit.> allow to coarsely describe the combinatorial structure of the interval [1,w]^W.


    The elements u ∈ [1,w]^W are split into 3 rows according to the following cases (where v is the right complement of u):
    
    	
  * (bottom row) u is horizontal elliptic and v is hyperbolic;
    	
  * (middle row) both u and v are vertical elliptic;
    	
  * (top row) u is hyperbolic and v is horizontal elliptic.
    
    The bottom and the top rows contain a finite number of elements, whereas the middle row contains infinitely many elements.


This coarse structure has the following implications, given elements u ≤ v in [1,w]^W: if v is elliptic, then u is elliptic; if v is horizontal elliptic, then u is horizontal elliptic; if u is vertical, then v is vertical; if u is hyperbolic, then v is hyperbolic.










The roots corresponding to horizontal reflections form a root system Φ_⊆Φ, called the horizontal root system associated with the Coxeter element w ∈ W.
It decomposes as a disjoint union of orthogonal irreducible root systems of type A, as described in <Ref>.
The number k of irreducible components varies from 1 to 3.
See <cit.> and <Ref>.







	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	The interval [1,w]^W is a lattice (and thus W_w is a Garside group) if and only if the horizontal root system associated with w is irreducible.
	This happens in the cases C̃_n, G̃_2, and Ã_n if w is a (n,1)-bigon Coxeter element.



Since the interval [1,w]^W is not a lattice in general, in <cit.> a new group of isometries C ⊇ W is constructed, with the property that [1,w]^C is a balanced lattice and [1,w]^W ⊆ [1,w]^C.
The corresponding interval group C_w (called braided crystallographic group) is a Garside group, and there is a natural inclusion W_w ⊆ C_w.

By <Ref>, the interval complex K_C associated with [1,w]^C is a (finite-dimensional) classifying space for C_w.
The cover of K_C corresponding to the subgroup W_w is a classifying space for the (dual) affine Artin group W_w.
Therefore affine Artin groups admit a finite-dimensional classifying space.
We come back to braided crystallographic groups in <Ref>, where we show that the interval complex K_W associated with [1,w]^W is a classifying space for W_w (this complex is much simpler than the aforementioned cover of K_C).

In the subsequent sections, we sometimes suppress the superscript W when writing intervals [1,u]^W in a Coxeter group W, and simply write [1,u].





§ AFFINE COXETER ELEMENTS



This section is devoted to proving some results on the geometry of Coxeter elements of affine Coxeter groups, expanding the theory of <cit.>.
We start by recalling, in <Ref>, the results of <cit.> on bipartite Coxeter elements.
In <Ref> we develop a parallel theory for the case Ã_n.
In <Ref> we prove a few structural results for the elements of the interval [1,w].
Finally, in <Ref> we make a digression on the geometry of the irreducible horizontal components.

This section goes hand in hand with the appendix, where we carry out explicit computations for the four infinite families of irreducible affine Coxeter groups.
A few results of <Ref> and one technical lemma of <Ref> are checked by hand in the appendix.
The appendix can be also used as a source of additional examples.

Let W be an irreducible affine Coxeter group, acting faithfully by Euclidean isometries on E = ^n where n is the rank of W, as described in <Ref>.
Let R ⊆ W be the set of reflections, w a Coxeter element of W, and ℓ = (w) ⊆ E the Coxeter axis of w.
Denote by  the reflection arrangement associated with the action of W on E.

The shortest vector μ in (w) gives an orientation to the Coxeter axis ℓ: we say that a point ∈ℓ is above a point ∈̱ℓ (or, equivalently, $̱ is below) if-$̱ is a positive multiple of μ.
We also say that μ points towards the positive direction of ℓ, whereas -μ points towards the negative direction.





 §.§ Bipartite Coxeter elements





	A bipartite Coxeter element is a Coxeter element w ∈ W for which there exist a set of simple reflections S ⊆ R and a bipartition S = S_0 ⊔ S_1 of the Coxeter graph (i.e. the reflections in S_0 pairwise commute, and so do the reflections in S_1) such that w = w_1w_0, where w_i is the product of the elements of S_i in some order.
	


If the Coxeter graph of W is a tree, then every Coxeter element w ∈ W is a bipartite Coxeter element <cit.>.
In particular, this happens for all irreducible affine Coxeter groups except Ã_n.

Let w be a bipartite Coxeter element, as in <Ref>.
Let C_0 be the (open) chamber of the Coxeter complex corresponding to the set S of simple reflections so that the elements of S = S_0 ⊔ S_1 are the reflections with respect to the walls of C_0.
Let F_i be the face of C_0 determined by the intersection of the hyperplanes of the reflections in S_i, and let B_i be the affine span of F_i.
There is a unique pair of points _i ∈ B_i that realize the minimum distance between B_0 and B_1.
Each _i lies in the relative interior of the corresponding face F_i <cit.>.
The line determined by _0 and _1 is exactly the Coxeter axis ℓ <cit.>, and it intersects C_0 <cit.>.

Each w_i is an involution, and it restricts to a reflection on the Coxeter axis ℓ that fixes only _i <cit.>.
Then w_0 and w_1 generate an infinite dihedral group that acts on ℓ.
Using this action, we can extend the definitions of F_i, B_i, and _i to arbitrary subscripts i ∈: let F_-i (resp. B_-i, or _-i) be the image of F_i (resp. B_i, or _i) under w_0, and let F_2-i (resp. B_2-i, or _2-i) be the image of F_i (resp. B_i, or _i) under w_1.
We obtain a sequence of equally spaced points _i along the line ℓ, with p_i+1 - p_i = 1/2μ (where μ is the shortest vector in (w)).
The axial chambers are given by all possible images of the chamber C_0 under this dihedral group action.
Their vertices are called axial vertices.


	
	If b is an axial vertex in the face F_i (for some i ∈), then w^j(b) is an axial vertex in F_i+2j.
	Therefore every axial chamber has exactly one vertex in the orbit { w^j(b) | j ∈}.




	Let W be an irreducible affine Coxeter group not of type Ã_n, and w one of its Coxeter elements.
	For every axial chamber C there is a bipartite factorization w = w_+w_-, where w_+ (resp. w_-) is the product of the reflections with respect to the walls of C that intersect ℓ above (resp. below) C.
	



If a hyperplane H of a reflection of W crosses the Coxeter axis ℓ, then there is an index i such that H contains all of F_i, all but one vertex of F_i-1 and all but one vertex of F_i+1 <cit.>.



	Let W be an irreducible affine Coxeter group not of type Ã_n, and w one of its Coxeter elements.
	Let H be the hyperplane of a vertical reflection r in W that intersects the Coxeter axis ℓ at the point _i.
	If $̱ and'̱are the unique vertices ofF_i-1andF_i+1not contained inH, thenwsends$̱ to '̱, r swaps $̱ and'̱,rwfixes$̱, and wr fixes '̱.
	Moreover, the elliptic isometry rw (resp. wr) is a Coxeter element for the finite parabolic subgroup of W that fixes $̱ (resp.'̱).
	
	Let W be an irreducible affine Coxeter group not of type Ã_n, and w one of its Coxeter elements.
	Every vertical reflection r ∈ W is in [1,w], and fixes many axial vertices.
	A horizontal reflection r ∈ W is in [1,w] if and only if it fixes at least one axial vertex.
[Case G̃_2]
	<Ref> shows the Coxeter complex of a Coxeter group of type G̃_2.
	Every Coxeter element w is a glide reflection, whose glide axis is the Coxeter axis ℓ = (w) (the dashed line).
	The axial chambers are shaded, and the axial vertices are colored, with one color for each w-orbit (see <Ref>).
	We use the notation of <cit.> to indicate the reflections in R_0 = R ∩ [1,w]: let C_0 be the darkly shaded chamber in <Ref>; let p_0 ∈ℓ be below C_0, and p_1 ∈ℓ above C_0; denote by a_i (for i ≡ 1 mod 4) the reflection with respect to the line of slope -√(3) passing through the point p_i ∈ℓ;
	similarly, denote by b_j, c_k, d_i, e_j the vertical reflections with slopes -1/√(3), 0, 1/√(3), √(3), respectively (they are defined for i ≡ 1 mod 4, j ≡ 3 mod 4, and k ≡ 0 mod 2);
	finally, let f and f' be the two horizontal reflections of [1,w].
	The walls of C_0 are the fixed lines of a_1, d_1, c_0.
	A bipartite factorization of w is w=a_1d_1c_0.
	

 §.§ Coxeter elements of type tilde An


IfWis a Coxeter group of typeÃ_n, most of its Coxeter elements are not bipartite, and thus the theory of <Ref> does not apply.
In this section, we derive a parallel theory and highlight the most important differences with the bipartite case.

As shown in <cit.>, every Coxeter element is geometrically equivalent to a (p,q)-bigon Coxeter element, for a unique pair(p,q)of positive integers such thatp ≥qandp+q = n+1.
Therefore there are exactly⌊n+1/2 ⌋distinct equivalence classes of Coxeter elements.
For the explicit construction of(p,q)-bigon Coxeter elements, see Section <ref> in the appendix.
The first four results in this section (<Ref>, <Ref>, <Ref>) are verified in the appendix by explicit computation.


	Let W be a Coxeter group of type Ã_n, and w one of its (p,q)-bigon Coxeter elements.
	The Coxeter axis ℓ is not contained in any reflection hyperplane of W, and it intersects the vertical hyperplanes in an infinite sequence of equally spaced points {p_i}_i ∈.
	More precisely we have p_i+1 - p_i = (p,q)/p+qμ, where μ is the shortest vector of (w).
	In particular, w(p_i) = p_j with j = i + p+q/(p,q).


As in the bipartite case, the chambers that intersect the Coxeter axisℓare called axial chambers, and the vertices of the axial chambers are called axial vertices.
The following theorem is the analog of <Ref>, and describes how axial chambers yield a factorization ofw.


    Let W be a Coxeter group of type Ã_n, and w one of its (p,q)-bigon Coxeter elements with p ≥ q.
    Fix an axial chamber C, and let S_C ⊆ R be the set of the n+1 reflections with respect to the walls of C.
    Write S_C = S^+ ⊔ S^- ⊔ S^, where S^+ (resp. S^-) consists of the reflections that intersect the Coxeter axis ℓ above (resp. below) C ∩ℓ, and S^ consists of the horizontal reflections.
    Then:
    
  * |S^+| = |S^-| = q, and |S^| = p-q;
        
  *  the reflections in S^+ (resp. S^-) pairwise commute;
        
  * w can be written as a product of the reflections in S_C, where the reflections in S^+ come first, and the reflections in S^- come last.
    


The next result describes how the vertical hyperplanes of the arrangementintersect the Coxeter axisℓ.
Unlike the bipartite case, the vertical walls of an axial chamberCcan intersectℓoutside of its closureC̅.


	Let W be a Coxeter group of type Ã_n, and w one of its (p,q)-bigon Coxeter elements with p ≥ q.
	
  *  Let ∈ℓ be a point which is fixed by at least one vertical reflection of W.
		There are exactly (p,q) vertical reflections of W that fix , and they pairwise commute.
		
		
  *  Let {p_i}_i ∈ be the sequence of points of <Ref>, and let C be an axial chamber that intersects ℓ between p_i and p_i+1.
		A vertical hyperplane of  is a wall of C if and only if it intersects ℓ in one of the 2m consecutive points p_i-m+1, p_i-m+2, …, p_i+m, where m = q/(p,q).
	

The following result gives some insight into the geometry of axial vertices.
The first part is theÃ_nanalog of <Ref>.


	Let W be a Coxeter group of type Ã_n, and w one of its (p,q)-bigon Coxeter elements.
	Let b be an axial vertex.
	
  *  Every axial chamber has exactly one vertex in the set { w^j(b) | j ∈}.
		
  *  There are exactly p+q/(p,q) axial chambers having b as one of their vertices, and they are consecutive (i.e. the union of their closures intersects the Coxeter axis ℓ in a connected set).
	
	If n is odd and p=q=n+1/2, then w is a bipartite Coxeter element, and we recover the results of <Ref>.
	In particular, part (iii) of <Ref> reduces to the bipartite factorization w = w_+w_- of <Ref>.
[Case Ã_2]
	<Ref> shows the Coxeter complex of a Coxeter group of type Ã_2.
	Every (2,1)-bigon Coxeter element w is a glide reflection, whose glide axis is the Coxeter axis ℓ = (w) (the dashed line).
	As in <Ref>, the axial chambers are shaded, and the axial vertices are colored, with one color for each w-orbit (see part (i) of <Ref>).
	We use a notation similar to <Ref>: let C_0 be the darkly shaded chamber; let p_0 ∈ℓ be the point immediately below C_0, and p_1 ∈ℓ the point immediately above C_0;
	denote by a_i the vertical reflection that fixes p_i for i ≡ 1 mod 2; denote by c_j the vertical reflection that fixes p_j for j ≡ 0 mod 2; finally, let b and b' be the two horizontal reflections adjacent to the Coxeter axis ℓ (by <Ref> below, these are precisely the horizontal reflections of [1,w]).
	The walls of C_0 are the fixed lines of a_1, b, c_0, and w = a_1bc_0.
	



IfHis the fixed hyperplane of some vertical reflection ofW, there is a well-defined axial chamber which is immediately aboveH, and one which is immediately belowH.
These are the only two chambers that intersect a small neighborhood ofH ∩ℓinℓ.
Denote them byC_H^+andC_H^-, respectively.
By part (i) of <Ref>, all the vertical reflections fixingH ∩ℓpairwise commute.
Therefore they are all walls ofC_H^+andC_H^-.
This proves the following analog of <cit.>.


	Let W be a Coxeter group of type Ã_n, and w a Coxeter element of W.
	If the fixed hyperplane H of a reflection in W crosses the axis ℓ, then H is the affine span of a facet of an axial chamber.













The following is theÃ_nanalog of <Ref>.


	Let W be a Coxeter group of type Ã_n, and w one of its Coxeter elements.
	Let r be a vertical reflection in W, H = (r), and ^̱- (resp. ^̱+) the unique vertex of C_H^- (resp. C_H^+) which is not in H.
	Then the elliptic isometry rw (resp. wr) is a Coxeter element for the finite parabolic subgroup of W that fixes ^̱- (resp. ^̱+).

	Apply <Ref> to the axial chamber C = C_H^-.
	Since r ∈ S^+, the Coxeter element w can be written as a product rr_2… r_n+1, where r_2, …, r_n+1 are the reflections with respect to the other walls H_2, …, H_n+1 of C_H^-.
	The walls H_2, …, H_n+1 bound a chamber of the finite parabolic subgroup of W that fixes ^̱-, so rw = r_2… r_n+1 is a Coxeter element for this subgroup.
	Similarly, wr is a Coxeter element for the finite parabolic subgroup that fixes ^̱+.

	In the proof of <Ref>, the chamber C_H^- (resp. C_H^+) can be replaced with any axial chamber C that intersects ℓ below (resp. above) H ∩ℓ and such that H is a wall of C.
	However, the statement of <Ref> makes it clear that at least one such chamber C exists, so rw and wr are indeed parabolic Coxeter elements.

	Let W be a Coxeter group of type Ã_n, and w one of its Coxeter elements.
	For every axial vertex $̱, there exists an axial chamberCsuch that$̱ is a vertex of C and the wall of C opposite to $̱ is vertical.
	

	Let C be the lowest axial chamber such that $̱ is a vertex ofC.
	Suppose by contradiction that the wall ofCopposite to$̱ is horizontal.
	Then $̱ is fixed by all the reflections with respect to the vertical walls ofC.
	Letbe the lowest point ofC̅∩ℓ, and letS_be the set of the (vertical) reflections ofWthat fix.
	By part (i) of <Ref>, the reflections inS_pairwise commute.
	Then their product fixes$̱ and sends C to an axial chamber C' which is below C.
	This is a contradiction.


We are now ready to prove theÃ_nanalog of <Ref>.


	Let W be a Coxeter group of type Ã_n, and w one of its Coxeter elements.
	Every vertical reflection r ∈ W is in [1,w], and fixes many axial vertices.
	A horizontal reflection r ∈ W is in [1,w] if and only if it fixes at least one axial vertex.

	For the first part, it is enough to apply <Ref> to C_H^+ or C_H^-, where H = (r).
	
	For the second part, suppose that r is a horizontal reflection in [1,w], so there is a factorization w = r_1r_2… r_n r that ends with r.
	Since w is vertical, at least one of the r_i is vertical.
	By <Ref> we can move this reflection to the beginning, and thus assume that r_1 is vertical.
	Then r ≤ w' = r_1w, and w' is an elliptic isometry that fixes an axial vertex ^̱- by <Ref>.
	Since (w') ⊆(r) (<Ref>), we have that r fixes ^̱-.
	
	On the other hand, suppose that r ∈ W is a (horizontal) reflection that fixes some axial vertex $̱.
	By <Ref>, we have that$̱ is a vertex of some axial chamber C such that the wall H' of C opposite to $̱ is vertical.
	Letr'be the reflection with respect toH'.
	By <Ref> and <Ref>, one ofr'wandwr'is a Coxeter element for the finite parabolic subgroup ofWthat fixes$̱.
	Recall that every reflection in a finite Coxeter group occurs in some minimal length factorization of any of its Coxeter elements <cit.>.
	Therefore r ≤ r'w or r ≤ wr', and so r ≤ w.


 §.§ Isometries below an affine Coxeter element


Now that the caseÃ_nis well understood, we turn to the general case of an irreducible affine Coxeter group and prove a few more results about the elements of the interval[1,w].



	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	For every axial vertex $̱, there exists a unique elementw_∈̱[1,w]which is a Coxeter element for the finite parabolic subgroup ofWthat fixes$̱.

	Let C be an axial chamber such that $̱ is a vertex ofC.
	Denote byr, r_1,…,r_nthe reflections with respect to the walls ofC, whereris the reflection that does not fix$̱.
	By <Ref> (for the case Ã_n) and <Ref> (for the other cases), the Coxeter element w can be written as a product of the reflections r, r_1,…,r_n in some order.
	Remove r from this factorization, and let w_$̱ be the product of the remaining reflections in the same relative order.
    Thenw_∈̱[1, w]by <Ref>.
	By construction,w_$̱ is a Coxeter element for the finite parabolic subgroup of W that fixes $̱.
	Every such elementw_'̱∈ [1,w]has a fix-set equal to{}̱, so uniqueness follows from <Ref>.

	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	For every elliptic element u ∈ [1,w], there exists an axial vertex $̱ such thatu ≤ w_$̱, where w_$̱ is the unique element of[1,w]which is a Coxeter element for the finite parabolic subgroup that fixes$̱ (see <Ref>).
	In particular, u fixes at least one axial vertex $̱.
	
	Let v be the right complement of u, so that uv = w.
	Let v = r_1… r_m be a minimal length factorization of v as a product of reflections.
	Since u is elliptic, v is vertical and therefore at least one r_i is a vertical reflection.
	By <Ref> we can move this reflection to the end, and thus assume that r_m is vertical.
	By <Ref> (for the case Ã_n) and <Ref> (for the other cases), wr_m = ur_1… r_m-1 is a  Coxeter element for the finite parabolic subgroup of W that fixes an axial vertex $̱.

	By construction,u ≤ w r_m.



Given an elementu ∈ [1,w], denote byW_uthe subgroup ofWgenerated by the reflections in[1,u](or, equivalently, by all the elements of[1,u]).
ThenW_uis a Coxeter group havingR ∩ W_uas its set of reflections, by <cit.> (see also <cit.>).
Denote by_u = {(r) | r ∈ R ∩ W_u }⊆the reflection arrangement associated with the Coxeter groupW_u.



	Let W be an irreducible affine Coxeter group, w one of its Coxeter elements, and u ∈ [1,w] a hyperbolic element.
	There exists a unique decomposition u=u'h such that:
	
  * u', h ∈ [1,u], u' is hyperbolic, h is horizontal elliptic, and l(u) = l(u') + l(h);
		
  * W_u is the direct product of the Coxeter subgroups W_u' and W_h;
		
  * W_u' is an irreducible affine Coxeter subgroup;
		
  * W_h is a finite horizontal Coxeter subgroup;
		
  * [1,u] = [1,u'] × [1,h].
	
	Decompose the Coxeter group W_u as a direct product of irreducible subgroups: W_u = W_1 ×…× W_t.
	Then u ∈ W_u can be written uniquely as u = u_1… u_t with u_i ∈ W_i.
	
	Since u is hyperbolic, its right complement v is horizontal elliptic.
	Let u = r_1… r_m be a minimal length factorization of u as a product of reflections.
	Since u is vertical, at least one r_i is a vertical reflection.
	By <Ref> we can move this reflection to the beginning, and thus assume that r_1 is vertical.
	Therefore its right complement r_2… r_m v is vertical elliptic, and in particular r_2… r_m is elliptic.
	Each reflection r_i belongs to one of the irreducible components W_j.
	Without loss of generality, assume that r_1, …, r_k ∈ W_1 and r_k+1, …, r_m ∈ W_2 ×…× W_t, for some k ∈{1, …, m}.
	By uniqueness of the decomposition u=u_1… u_t, we have that u_1 = r_1… r_k and u_2… u_t = r_k+1… r_m.
	In particular, both u_1 and u_2… u_t belong to the interval [1,u], and l(u) = l(u_1) + l(u_2… u_t) = l(u_1) + l(u_2) + … + l(u_t).
	
	Denote by α_1, …, α_m the roots corresponding to r_1, …, r_m.
	Since u_2… u_t = r_k+1… r_m is elliptic, the roots α_k+1, …, α_m are linearly independent by <cit.>.
	Suppose by contradiction that u_1 = r_1… r_k is also elliptic. Then for the same reason the roots α_1,…, α_k are linearly independent.
	Therefore the roots α_1,…, α_m are linearly independent, so u is elliptic by <cit.>, and this is a contradiction.
	We deduce that u_1 is hyperbolic.
	This implies that its right complement r_k+1… r_m v is horizontal elliptic, so the reflections r_k+1, …, r_m are horizontal.
	Then u_2… u_t = r_k+1… r_m is horizontal elliptic.
	
	Recall that W_u is generated by the reflections in [1,u].
	Each reflection r ∈ [1,u] belongs to exactly one irreducible factor W_i, and so is part of a minimal length factorization of u_i, which implies that r ∈ [1,u_i].
	Therefore W_i is equal to the subgroup W_u_i⊆ W generated by the reflections in [1,u_i].
	For all i ≥ 2 we have that u_i is horizontal elliptic, so all the elements of W_u_i are horizontal.
	Then the irreducible factor W_1 = W_u_1 is uniquely determined as the only factor which contains at least one vertical reflection.
	
	Define u' = u_1 and h = u_2… u_t.
	Notice that W_h is the group generated by the reflections in [1,h], i.e. the reflections in [1,u_2] ∪…∪ [1,u_t], so W_h = W_u_2×…× W_u_t = W_2 ×…× W_t.
	Therefore W_u = W_u'× W_h.
	This, together with the fact that W_h is horizontal and W_u' is irreducible, is enough to ensure that the decomposition u = u'h is unique.
	Since W_u = W_u'× W_h, we have [1,u] = [1,u'] × [1,h].

	
	Recall that u' = r_1… r_k is hyperbolic, so (r_1) ∩…∩(r_k) = ∅.
	Therefore the Coxeter subgroup W_u' is infinite.
	By construction, W_u' is also irreducible, so it must be an irreducible affine Coxeter group.


We will refer to the decompositionu = u'hof <Ref> as the hyperbolic-horizontal decomposition ofu.

The following technical lemma is proved in the appendix for the four infinite families and was checked by computer for the exceptional cases (see <cit.>).


	Let W be an irreducible affine Coxeter group, w one of its Coxeter elements, and u ∈ [1,w] a hyperbolic element such that W_u is irreducible.
	Let  be a point of ℓ that does not lie on any hyperplane of _u, and let C be the chamber of _u containing .
	Then C has exactly l(u) walls, and u can be written as the product of the reflections with respect to the walls of C in the following order:
	
  *  first there are the vertical reflections that fix a point of ℓ above , and r comes before r' if (r) ∩ℓ is below (r') ∩ℓ;
		
  *  then there are the horizontal reflections, in some order;
		
  *  finally there are the vertical reflections that fix a point of ℓ below , and again r comes before r' if (r) ∩ℓ is below (r') ∩ℓ.
	

The conclusion of <Ref> seems to hold for all hyperbolic elementsu ∈ [1,w], without the irreducibility hypothesis.
In addition, for the caseG̃_2and for the four infinite families (Ã_n,B̃_n,C̃_n, andD̃_n), the vertical walls ofCthat fix a point ofℓabove(resp. below) pairwise commute.
However, a computer check shows that this is not true for the exceptional casesF̃_4,Ẽ_6,Ẽ_7, andẼ_8(see <cit.>).
Notice also that in general(w) ⊈(u), so a point∈ℓ = (w)is usually not on the Coxeter axis ofu(otherwise <Ref> would follow easily from <Ref> applied toW_u).

We can now prove the affine analog of <cit.>.


	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	Every element u ∈ [1,w] is a Coxeter element of W_u.
	In addition:
	
  *  if u is elliptic, then u is a parabolic Coxeter element;
		
  *  if u is hyperbolic and u=u'h is its hyperbolic-horizontal decomposition, then u' is a Coxeter element of W_u' and h is a Coxeter element of the parabolic subgroup W_h of W_u.
	
	Suppose that u is elliptic.
	By <Ref> there is an axial vertex $̱ such thatu ≤ w_$̱, where w_$̱ is the parabolic Coxeter element of <Ref>.
	By <cit.> applied to the finite parabolic subgroup that fixes$̱, we have that u is a Coxeter element of W_u, and W_u is a parabolic subgroup of W.

	Suppose now that u is hyperbolic, and let u=u'h be its hyperbolic-horizontal decomposition.
	Since W_u' is irreducible, by <Ref> we immediately have that u' is a Coxeter element of W_u'.
	Also, since h is horizontal, we have already proved that h is a Coxeter element of the parabolic subgroup W_h (notice that W_h is a parabolic subgroup of both W and W_u).
	Then u'h is a Coxeter element of W_u = W_u'× W_h.


In the caseC̃_n, the first part of <Ref> was already noted by Digne <cit.>.



	If W is an irreducible affine Coxeter group, then all its proper parabolic subgroups are finite.
	Therefore, if u ≠ w is a hyperbolic element, then W_u is not a parabolic subgroup (and u is not a parabolic Coxeter element).
[Hyperbolic elements in the case C̃_3]
	If W is a Coxeter group of type C̃_3, the interval [1,w] has 3 hyperbolic elements of length 2 (the translations), 6 of length 3, and 1 of length 4 (w itself).
	They are the complements of the horizontal elements, which are explicitly described in <Ref> below.
	Among the 6 hyperbolic elements u ∈ [1,w] of length 3, in 3 cases W_u ≅ W_Ã_1× W_A_1 (so the hyperbolic-horizontal decomposition of u has a non-trivial horizontal factor), whereas in the remaining three cases W_u ≅ W_C̃_2.
	See Section <ref> for an explicit computation of the hyperbolic elements in the case C̃_n.
	

 §.§ Horizontal components



We now describe the geometry of the irreducible horizontal components of an affine Coxeter group.
The ideas for this section are mostly already present in <cit.>, but we find it convenient to write them down more explicitly.

As in <Ref>, letWbe an irreducible affine Coxeter group acting onE = ^nwherenis the rank ofW, and fix a Coxeter elementw.
LetΦbe the root system ofW, and letΦ_⊆Φbe the horizontal root system.

As shown in <cit.>, there exists at least one horizontal factorizationw = thwheret ∈ [1,w]is a translation,h ∈ [1,w]is a horizontal isometry of reflection lengthn-1, and every horizontal reflection ofWis parallel to some reflection of the parabolic Coxeter subgroupW_h.
In other words, the root system ofW_hcoincides with the horizontal root systemΦ_ofW.

The horizontal root systemΦ_decomposes as a disjoint union of orthogonal irreducible root systems of typeA<cit.>:Φ_ = Φ_1⊔Φ_2⊔…⊔Φ_k, whereΦ_iis a root system of typeA_n_i, andn_1+n_2+…+n_k = n-1(see <Ref>).
Accordingly, the horizontal isometryhdecomposes as a producth = h_1h_2… h_kwhereh_ibelongs to thei-th irreducible horizontal component and has reflection lengthn_i.
ThenΦ_iis the root system of the irreducible parabolic Coxeter subgroupW_h_i, andh_iis a Coxeter element ofW_h_iby <Ref>.

We now focus on a single irreducible horizontal componentΦ_iand describe the geometry of the associated horizontal reflections.
Letm = n_ibe the rank ofΦ_i.
SinceΦ_iis a root system of typeA_m, the reflections ofWin the directions ofΦ_igenerate a Coxeter subgroupW_i ⊆ Wof typeÃ_m.
Denote by_ithe hyperplane arrangement associated withW_i(this is a subarrangement of the hyperplane arrangementassociated withW).

The Coxeter axisℓdoes not intersect any horizontal hyperplane, and so it is contained in one chamberC_iof the arrangement_i.
We callC_ithei-th horizontal prism.
Like every chamber of_i, it hasm+1faces of minimal dimensionn-m.
We call them the minimal faces ofC_i.


Since the Coxeter axisℓis contained inC_i, every axial chamber ofis also contained inC_i.
As a consequence, all axial vertices are contained inC̅_i.
The horizontal isometryh_ihas reflection lengthm, so(h_i) = n-m, and(h_i)is the intersection ofmhyperplanes of_i.
Sinceh_i ∈ [1,w], by <Ref> we have that(h_i)contains at least one axial vertex.
Therefore(h_i)is one of the minimal faces ofC_i.



	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	Every horizontal prism C_i is fixed by w (as a set).
	In addition, w acts transitively on the set of walls of C_i and on the set of minimal faces of C_i.

	From the factorization w=th_1h_2 … h_k we see that w sends hyperplanes of _i to hyperplanes of _i.
	Also, it fixes the Coxeter axis ℓ (as a set).
	Therefore it fixes C_i (as a set), it permutes the walls of C_i, and it permutes the minimal faces of C_i.
	The action of w on the walls of C_i is the same as the action of th_i, because h_j fixes every hyperplane of _i for all j ≠ i.
	
	Let H_1, …, H_m+1 be the walls of C_i, and for every j ∈{1, …, m+1} let α_j be the root of H_j that points from H_j towards the half-space containing C_i.
	The linear part of th_i (which coincides with the linear part of h_i) permutes α_1, …, α_m+1.
	Recall that h_i is a Coxeter element of W_h_i, which is a Coxeter group of type A_m with root system Φ_i.
	In a finite Coxeter group, the orbits of roots under the action of a Coxeter element are known to have cardinality equal to the Coxeter number, which is m+1 for a Coxeter group of type A_m.
	Therefore h_i transitively permutes the roots α_1, …, α_m+1.
	Then w transitively permutes the walls of C_i.
	Every minimal face of C_i is opposite to exactly one wall of C_i, so the same conclusion holds for the minimal faces.


Letφ [1,w] → [1,w]be the conjugation by the Coxeter elementw:φ(u) = w^-1uw.


	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	Every minimal face of the i-th horizontal prism C_i is the fixed set of φ^p(h_i) for some p ∈{0, …, m}, and it contains at least one axial vertex.
	In addition, the elements φ^p(h_i) for p ∈{0, …, m} are the maximal elements of the subposet [1,w] ∩ W_i ⊆ [1,w], and they all have the same linear part.

	We have that (h_i) is a minimal face of C_i.
	Then the first part follows from <Ref>.
	Since φ^p(h_i) ∈ [1,w], its fixed set contains at least one axial vertex by <Ref>.
	
	Every element of [1,w] ∩ W_i of reflection length m must have a fixed set equal to a minimal face of C_i.
	By <Ref>, there can be at most one such element for every minimal face of C_i.
	Therefore, the elements φ^p(h_i) for p ∈{0, …, m} are the only elements of reflection length m in [1,w] ∩ W_i.
	Every other element u ∈ [1,w] ∩ W_i has a reflection length strictly smaller than m.

	We want to show that, for every u ∈ [1,w] ∩ W_i, we have u ≤φ^p(h_i) for some p.
	Let v be the left complement of u, so that vu = w.
	Then v is hyperbolic.
	Let v = v'h' be the hyperbolic-horizontal decomposition of v (see <Ref>).
	Recall that the Coxeter subgroup W_v' generated by [1,v'] is an irreducible affine Coxeter group, and v' is one of its Coxeter elements by <Ref>.
	Then the interval [1,v'] contains at least one translation t' (this follows for example from the existence of a horizontal factorization of v' in W_v').
	Therefore we can write v' = t'v”, with l(v') = l(t') + l(v”).
	Putting everything together, we get the factorization w = t'v”h'u, with l(t') + l(v”) + l(h') + l(u) = l(w) = n+1.
	Since t' is a translation, and so has length l(t') = 2, its right complement h̅ = v”h'u is a horizontal element of length l(h̅) = n-1.
	In addition, u ≤h̅ because u is part of a minimal length factorization of h̅.
	Write h̅ = h̅_1 h̅_2 …h̅_k, with h̅_j ∈  [1,w] ∩ W_j.
	We have l(h̅_j) ≤ n_j for all j, and n_1 + … + n_k = n-1, so l(h̅_j) = n_j.
	In particular l(h̅_i) = n_i = m, so h̅_i = φ^p(h_i) for some p ∈{0, …, m}.
	Since u ≤h̅ and u ∈ [1,w] ∩ W_i, we have u ≤h̅_i = φ^p(h_i).
	
	Finally, we want to show that the linear part of φ^p(h_i) is equal to the linear part of h_i.
	This follows from the factorization w = th_1h_2… h_k, together with the fact that h_j commutes with h_i for all j ≠ i, and that the linear part of the translation t is the identity.

	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	For every horizontal root α∈Φ_, the interval [1,w] contains exactly two reflections in the direction of α, namely those determined by adjacent hyperplanes which contain the Coxeter axis ℓ between them.

	Let Φ_i be the irreducible component of Φ_ containing α.
	A hyperplane of _i yields a reflection in [1,w] if and only if it contains an axial vertex, which happens if and only if it contains a minimal face of C_i (by <Ref>).
	In the arrangement _i, which is of type Ã_m, there are exactly two hyperplanes in the direction of α that contain at least one minimal face of C_i.
	These two hyperplanes are adjacent, and they contain C_i (and therefore also the Coxeter axis ℓ) between them.

	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	There exists an integer p>0 such that w^p is a translation in the positive direction of the Coxeter axis ℓ.

	Let n_1, …, n_k be the ranks of the irreducible components of the horizontal root system Φ_.
	If p is a multiple of n_i+1 for every i, then w^p acts trivially on all horizontal directions, and so it must be a translation in the direction of the Coxeter axis ℓ.
	If μ is the shortest vector in (w), then pμ∈(w^p). Therefore w^p is a translation of pμ, which is in the positive direction of ℓ.

	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	For every irreducible component Φ_i of the horizontal root system Φ_, there exists a hyperbolic element u ∈ [1,w] such that W_u is an irreducible affine Coxeter group with horizontal root system Φ_i (with respect to the Coxeter element u).
	In particular, [1,u] ∩ W_i = [1,w] ∩ W_i.
	
	Let Φ_ = Φ_1 ⊔…⊔Φ_k.
	If k=1, then we can simply take u = w.
	Suppose from now on that k ≥ 2, and assume without loss of generality that i=1.
	Let w=th_1… h_k be a horizontal factorization, with h_j ∈ W_j.
	Notice that t does not commute with any h_j, because t^-1h_j t = φ(h_j) ≠ h_j (by <Ref>).
	Let u = th_1 ≤ w.
	
	Since h_2… h_k is horizontal, its left complement u is hyperbolic.
	Let u = u'h' be the hyperbolic-horizontal decomposition of u (see <Ref>).
	The irreducible root system Φ_1 is entirely contained in the root system of W_u' or of W_h', because W_u = W_u'× W_h'.
	Since t is a (vertical) translation and [1,u] = [1,u'] × [1,h'], we have t ≤ u'.
	Thus Φ_1 is contained in the root system of W_u', because otherwise h_1 would commute with t.
	
	By <Ref> and <cit.> we have ((w)) ⊆((u')), so every root which is horizontal with respect to w is horizontal also with respect to u' ≤ w.
	Therefore the horizontal root system Φ_' of W_u' (associated with the Coxeter element u') contains Φ_1.
	On the other hand, l(u') ≤ l(u) = l(t) + l(h_1) = n_1 + 2, so the rank of Φ_' is at most n_1.
	We conclude that Φ_' = Φ_1 and u' = u.
	
	Finally we have [1,u] ∩ W_1 ⊆ [1,w] ∩ W_1, and every reflection in [1,w] ∩ W_1 is also contained in [1,u] by <Ref>, so actually [1,u] ∩ W_1 = [1,w] ∩ W_1.

	By <Ref>, in order to study the geometry of horizontal components it is enough to look at affine Coxeter groups with a unique horizontal component (Ã_n with a (n,1)-bigon Coxeter element, C̃_n, and G̃_2).
	The group G̃_2 has a horizontal component of rank 1.
	The groups Ã_n and C̃_n (see Sections <ref> and <ref> in the appendix) have a horizontal component of rank m = n-1.
	In all cases, choosing suitable coordinates we have that: the arrangement _i consists of the hyperplanes { x_j - x_j' = q } for 1 ≤ j < j' ≤ m and q ∈; the horizontal prism C_i is described by the inequalities x_1 < x_2 < … < x_m < x_1 + 1; the linear part of the Coxeter element w sends (x_1, …, x_m) to (x_m, x_1, …, x_m-1).
	In particular, the isomorphism type of [1,w] ∩ W_i (as a labeled poset) depends only on the rank m, and not on the ambient group W.
	Using ideas from the proofs of <cit.>, one can show that [1,w] ∩ W_i is isomorphic to the subposet of the noncrossing partition lattice of type B_m+1 consisting of the partitions without a zero block (this terminology is defined for example in <cit.>).
	[Horizontal component of rank 2]
	<Ref> shows the arrangement _i of a horizontal component of rank 2.
	The horizontal prism C_i is shaded, and its minimal faces are the three white vertices.
	The reflections in [1,w] ∩ W_i are denoted by a,a',b,b',c,c', and they correspond to the 6 thick lines.
	The Coxeter axis ℓ is contained in C_i and is equidistant from the minimal faces.
	The Coxeter element w acts on _i as a 2π/3 rotation around ℓ, say counterclockwise.
	Then the 3 maximal elements of [1,w] ∩ W_i are ab=bc=ca, a'b = bc' = c'a', and ab' = b'c' = c'a.
	They are 2π/3 rotations around the minimal faces of C_i, in counterclockwise direction.
	Since a'b ∈ [1,w] and a'b' ∉[1,w], the right complement u of a' is such that b ≤ u and b' ≰u in [1,w].
	This is an example where the converse of <Ref> does not hold: we have both b ≤ u and b' ≤ u in [1,w]^L, but b' ≰u in [1,w] = [1,w]^W.
	

§ SHELLABILITY OF AFFINE NONCROSSING PARTITION POSETS


In this section, we construct an EL-labeling for the noncrossing partition poset[1,w], wherewis any Coxeter element of an affine Coxeter groupW.
Therefore the poset[1,w]is EL-shellable.
This extends the analog result of Athanasiadis, Brady, and Watt for noncrossing partition lattices associated with finite Coxeter groups <cit.>.

The EL-labelings of finite noncrossing partition lattices play a fundamental role in our construction and are recalled in <Ref>.
However, the need for a global labeling and the presence of hyperbolic intervals make the affine case substantially different from the finite case.

The EL-labeling of[1,w]is going to be used in <Ref>, to complete the proof of theK(π,1)conjecture.




 §.§ Reflection orderings and shellability of finite noncrossing partition lattices


We start by describing the EL-la­be­lings of <cit.> for the noncrossing partition lattices associated with finite crystallographic Coxeter groups.

LetWbe a finite Coxeter group acting onV = ^nby linear isometries, and letRbe its set of reflections.
Denote byΦ⊆ Vthe root system ofW, and letΦ^+ ⊆Φbe a positive root system.
Forα∈Φ^+, denote byr_α∈ Rthe orthogonal reflection with respect toα.


	A total ordering ≺ of R is called a reflection ordering for W if, whenever α, α_1, α_2 ∈Φ^+ are distinct positive roots and α is a positive linear combination of α_1 and α_2, we have either
	
    r_α_1≺ r_α≺ r_α_2  or   r_α_2≺ r_α≺ r_α_1.

	Let w be a Coxeter element of W.
	A reflection ordering ≺ of R is compatible with w if for every irreducible rank 2 induced subsystem Φ' ⊆Φ the following holds: if α and β are the simple roots of Φ' with respect to the positive system Φ' ∩Φ^+ and r_α r_β∈ [1,w], then r_α≺ r_β.


Letwbe a Coxeter element ofW.
Recall that the edges of the Hasse diagram of[1,w]are naturally labeled by reflections:λ(u, ur) = r.
We callλ([1,w]) → Rthe natural edge labeling of[1,w].


	Let W be a finite crystallographic Coxeter group, R its set of reflections, and w one of its Coxeter elements.
	If R is totally ordered by a reflection ordering which is compatible with w, then the natural edge labeling of [1,w] is an EL-labeling.


Notice that every finite Coxeter subgroup of an irreducible affine Coxeter group is crystallographic.
We are going to use <Ref> through the following geometric construction of reflection orderings, which is similar to a construction already considered in <cit.>.

Letbe the reflection arrangement associated withW, and letC_0be the chamber of the Coxeter complex ofWcorresponding to the choice of the positive systemΦ^+.
Fix a point∈ C_0and a non-zero vectorμ∈ V. Consider the affine lineℓ' = { + θμ|θ∈}⊆ V, with basepointa.
Assume thatℓ'is generic with respect to: it intersects every hyperplane ofin exactly one point (equivalently, it is not parallel to any hyperplane of), andH ∩ℓ' ≠ H' ∩ℓ'for all hyperplanesH ≠ H'in.
The vectorμgives an orientation toℓ'. In accordance with the notation of <Ref>, we say that a point∈̱ℓ'is above a point'̱∈ℓ'(or, equivalently,'̱is below$̱) if -̱'̱ is a positive multiple of μ.
Define a total ordering ≺_ℓ' on R as follows:

  *  first, there are the reflections that fix a point of ℓ' above , and r comes before r' if (r) ∩ℓ' is below (r') ∩ℓ';
	
  *  then there are the reflections that fix a point of ℓ' below , and again r comes before r' if (r) ∩ℓ' is below (r') ∩ℓ'.

Notice the similarity with <Ref>.



	Let W be a finite Coxeter group.
	For every generic line ℓ' as above, the total ordering ≺_ℓ' of R is a reflection ordering for W.

	Denote by ·,· the scalar product of V = ^n.
	By definition of the chamber C_0, we have that , α > 0 for every positive root α∈Φ^+.
	Until the end of this proof, renormalize all positive roots α∈Φ^+ so that , α = 1.
	
	Let α∈Φ^+.
	The intersection point + θ_αμ of (r_α) with ℓ' is determined by the relation + θ_αμ, α = 0.
	Since , α = 1, we get θ_α = - μ, α^-1.
	By definition of ≺_ℓ', we have that r_α≺_ℓ' r_β if and only if θ_α^-1 > θ_β^-1, which happens if and only if μ, α < μ, β.
	
	Now suppose that α∈Φ^+ is a positive linear combination of α_1, α_2 ∈Φ^+.
	Since , α =  , α_1 = , α_2  = 1, we have α = cα_1 + (1-c) α_2 with 0 < c < 1.
	Then μ, α = c μ, α_1  + (1-c) μ, α_2, which is between μ, α_1 and μ, α_2.
	Therefore r_α is between r_α_1 and r_α_2 in the total ordering ≺_ℓ'.


 §.§ Orderings of horizontal reflections


Let W be an irreducible affine Coxeter group, acting on an affine space E = ^n by Euclidean isometries, where n is the rank of W.
As usual, denote by R its set of reflections, and let w be one of its Coxeter elements.
Let Φ be the root system associated with W, and Φ_⊆Φ the horizontal root system.

Recall from <Ref> that the reflections occurring as labels of the interval [1,w] are those that fix at least one axial vertex.
This set R_0 ⊆ R includes the set R_$̌ of all vertical reflections, and a finite setR_of horizontal reflections, with two consecutive horizontal reflections for each pair of opposite roots.
We are going to construct a total order≺ofR_0which makes the natural edge labelingλ([1,w]) → R_0an EL-labeling of[1,w].
To do this, we start by defining a total ordering≺_on the subsetR_⊆ R_0of the horizontal reflections.

































We use the notation of <Ref>.
LetΦ_ = Φ_1 ⊔…⊔Φ_k, whereΦ_iis an irreducible root system of typeA_n_i.
Fix a horizontal factorizationw = th_1… h_k, so thatW_h_i⊆ Wis a finite parabolic subgroup with root systemΦ_i, andh_iis a Coxeter element ofW_h_i.
For now, we focus on a single horizontal componenti.
Letm = n_ibe the rank ofΦ_i, and letW_i ⊆ Wbe the Coxeter subgroup of typeÃ_mgenerated by the reflections with respect to roots inΦ_i.
Denote by_ithe corresponding hyperplane arrangement, and letC_ibe thei-th horizontal prism.


	Let u = φ^p(h_i) be any maximal element of [1,w] ∩ W_i, as in <Ref>.
	Fix any point a of the Coxeter axis ℓ.
	There exists a line ℓ', with basepoint a and direction in (Φ_i), such that the reflection ordering ≺_ℓ' for W_u (defined in <Ref>) is compatible with u.

	Choosing coordinates as in <Ref>, we can assume that: the arrangement _i consists of the hyperplanes { x_j - x_j' = q } for 1 ≤ j < j' ≤ m and q ∈; the horizontal prism C_i is described by the inequalities x_1 < x_2 < … < x_m < x_1 + 1; the element u sends (x_1, …, x_m) to (x_m, x_1, …, x_m-1), and the minimal face fixed by u is given by {x_1 = x_2 = … = x_m}.
	For a small enough ϵ > 0, the line ℓ' = { a + θ (1, ϵ, ϵ^2, …, ϵ^m-1) |θ∈} intersects the hyperplanes { x_j - x_j' = 0} (with j < j') in the lexicographic order of the pairs (j,j'), and always for θ > 0.
	This is the reflection ordering described in <cit.>, and it is compatible with the Coxeter element u.
	In order to get a line with direction in (Φ_i), project ℓ' to the affine subspace parallel to (Φ_i) and containing a.


By the previous lemma, there exists a reflection ordering≺_i'forW_h_iwhich is compatible with the Coxeter elementh_i.
Extend≺_i'to a total ordering≺_iof the setR_∩ W_i, in the following way: wheneverr_1 ≺_i' r_2inR_∩ W_h_i, then every parallel translate ofr_1comes before every parallel translate ofr_2.
Since there are no minimal factorizations ofwthat use two parallel horizontal reflections, the relative order of parallel horizontal reflections is not important and can be chosen arbitrarily.


	The total ordering ≺_i does not restrict to an ordering of <Ref> for u ≠ h_i.
	Rather, it is obtained by “translating” a chosen ordering for W_h_i (given by <Ref>) to the other subgroups W_u.
	There exists no total ordering of R_∩ W_i which restricts to an ordering of <Ref> for every maximal element u since the reflections with respect to the walls of C_i would necessarily form a loop.

	For every horizontal element u ∈ [1,w] ∩ W_i, the total ordering ≺_i makes the natural edge-labeling λ([1,u]) → R_∩ W_i an EL-labeling of [1,u].

	It is enough to prove this for the maximal elements of [1,w] ∩ W_i.
	By <Ref>, these are of the form u = φ^p(h_i) for p ∈{0, …, m}.
	
	Both (h_i) and (u) are minimal faces of the i-th horizontal prism C_i.
	Let t' be a translation that sends (h_i) to (u) (it does not need to be an element of W).
	The linear part of h_i is equal to the linear part of u by <Ref>, so t' h_i t'^-1 = u.
	Also, t' sends any hyperplane containing (h_i) to a parallel hyperplane containing (u).
	Therefore the conjugation by t' is an isomorphism W_h_i→ W_u that sends the Coxeter element h_i to the Coxeter element u, and it sends any reflection in W_h_i to its unique parallel translate in W_u.
	In particular, this isomorphism preserves the total ordering ≺_i.
	
	By construction, ≺_i restricts to a reflection ordering for W_h_i compatible with h_i, so the natural edge-labeling λ([1,h_i]) → R_∩ W_i is an EL-labeling of [1,h_i] by <Ref>.
	Using the above isomorphism W_h_i→ W_u, we obtain the same conclusion for u.
[Ordering of horizontal reflections in a component of rank 2]
	Consider a horizontal component of rank 2, with the notation of <Ref>.
	If we choose h_i = ab as the preferred maximal element of [1,w] ∩ W_i, one of the possible total orderings ≺_i of R_∩ W_i is the following: a ≺_i a' ≺_i c ≺_i c' ≺_i b ≺_i b'.
	The factorizations of the 3 maximal elements as a ≺_i-increasing product of reflections are: ab, a'b, and ab'.



Let≺_be any total ordering ofR_obtained as a shuffle of the total orderings≺_ifori ∈{1,…, k}.


	Let W be an irreducible Coxeter group, and w one of its Coxeter elements.
	For every horizontal element u ∈ [1,w], the total ordering ≺_ makes the natural edge labeling λ([1,u]) → R_ an EL-labeling of [1,u].

	Write u = u_1u_2… u_k, with u_i ∈ [1,w] ∩ W_i.
	Then [1,u] = [1,u_1] × [1,u_2] ×…× [1,u_k].
	We have an EL-labeling on each factor [1,u_i] by <Ref>, and we conclude using <Ref>.


 §.§ Axial orderings and shellability of affine noncrossing partition posets


We are now ready to construct an EL-labeling of the interval[1,w]in an affine Coxeter groupW, for any fixed Coxeter elementw.
As in <Ref>, we assume thatWis irreducible.
Once the irreducible case is settled, one can easily get an EL-labeling for reducible affine Coxeter groups by applying <Ref>.

Letℓbe the Coxeter axis, and fix an axial chamberC_0of the Coxeter complex.


	An axial ordering of the set of reflections R_0 = R ∩ [1,w] is a total ordering of the following form:
	
  *  first, there are the vertical reflections that fix a point of ℓ above C_0, and r comes before r' if (r) ∩ℓ is below (r') ∩ℓ (we call these the positive vertical reflections);
		
  *  then, there are the horizontal reflections in R_, following any total ordering ≺_ constructed in <Ref>;
		
  *  finally, there are the vertical reflections that fix a point of ℓ below C_0, and again r comes before r' if (r) ∩ℓ is below (r') ∩ℓ (we call these the negative vertical reflections).
	
	The relative order of vertical reflections that fix the same point of ℓ can be chosen arbitrarily.

	If two vertical reflections fix the same point of ℓ, they commute.
	This is proved in <Ref> for the case Ã_n, and follows from <Ref> for the other cases.
[Axial orderings for Ã_2 and G̃_2]
	In the case Ã_2, with the notation of <Ref>, one of the two axial orderings of R_0 is the following:
	
    a_1 ≺ c_2 ≺ a_3 ≺…≺ b ≺ b' ≺…≺ c_-2≺ a_-1≺ c_0.

	The other axial ordering is obtained by exchanging the two horizontal reflections b and b'.
	Notice that there are infinitely many reflections before b, and infinitely many reflections after b'.
	Indeed, b has no immediate predecessor, and b' has no immediate successor.
	The following is a portion of one of the infinitely many axial orderings in the case G̃_2, using the notation of <Ref>:
	
    a_1 ≺ d_1 ≺ c_2 ≺ e_3 ≺ b_3 ≺ c_4 ≺…≺ f ≺ f' ≺…≺ c_-2≺ e_-1≺ b_-1≺ c_0.



Our aim for the rest of this section is to prove that an axial ordering makes the natural edge labelingλ([1,w]) → R_0an EL-labeling of[1,w].



	Let [u,v] be an interval in [1,w].
	The map f  [1, u^-1v] → [u,v] defined by f(x) = ux is a label-preserving poset isomorphism.



Notice that an axial ordering ofR_0is not a well-ordering.
For example, the set of negative vertical reflections does not have a smallest element.
However, the well-ordering property holds for the subsets of the formR_0 ∩ [1,u], as we show in the second of the next three preparatory lemmas.


	For every hyperbolic element u ∈ [1,w], the interval [1,u] contains at least one positive vertical reflection and at least one negative vertical reflection.

	Since u is vertical, the interval [1,u] contains at least one vertical reflection r.
	Let w^p be a power of w that acts as a translation in the positive direction of the Coxeter axis ℓ, where p is a positive integer (see <Ref>).
	We have that w^p commutes with the right complement v of u, because v is horizontal.
	Then w^p commutes also with u = wv^-1.
	In particular, the conjugation by w^p fixes u and is an automorphism of [1,u].
	If we conjugate the vertical reflection r ∈ [1,u] by w^mp, for a sufficiently large m ∈, we get a vertical reflection r' = w^-mprw^mp∈ [1,u] that fixes a point of ℓ below C_0, i.e. r' is negative.
	Similarly, if we conjugate r by w^mp for a sufficiently small m ∈, we get a positive vertical reflection r”∈ [1,u].

	Let ≺ be an axial ordering of R_0.
	For every element u∈ [1,w] with u ≠ 1, the set R_0 ∩ [1,u] has a ≺-smallest reflection and a ≺-largest reflection.

	Suppose by contradiction that there is no smallest reflection in R_0 ∩ [1,u].
	Since R_0 ∩ [1,u] is non-empty, we must have an infinite decreasing sequence of reflections r_1 ≻ r_2 ≻… with r_i ∈ [1,u].
	In particular, the interval [1,u] is infinite, so u is hyperbolic.
	By <Ref>, there is at least one positive vertical reflection r ∈ [1,u].
	There is only a finite number of reflections r' ≼ r in R_0, so there exists a ≺-smallest reflection in R_0 ∩ [1,u].
	Similarly, there is also a ≺-largest reflection.










In the definition of an EL-labeling, maximal chains are compared lexicographically.
In our case it is also useful to compare them colexicographically (or antilexicographically): a tuple(r_1, r_2, …, r_m)is colexicographically smaller than(r_1', r_2', …, r_m')if the reflected tuple(r_m, r_m-1, …, r_1)is lexicographically smaller than the reflected tuple(r_m', r_m-1', …, r_1').



	Fix an axial ordering ≺ of R_0.
	Every interval [u,v] in [1,w] has a unique lexicographically smallest maximal chain, and this chain is increasing.
	Similarly, every interval [u,v] in [1,w] has a unique colexicographically largest maximal chain, and this chain is increasing.

	We proceed by induction on the length of the interval [u,v], the case u=v being trivial.
	Suppose from now on that u < v.
	The labels of the covering relations u ⋖ u' with u' ∈ [u,v] are all distinct, and they are given by the reflections in R_0 ∩ [1, u^-1v].
	By <Ref>, there is a ≺-smallest reflection r in R_0 ∩ [1, u^-1v].
	Let u' = ur.
	In view of the induction hypothesis applied to [u',v], it is enough to prove that all covering relations in [u',v] have labels greater than r.
	If r' is a label of some covering relation in [u',v], by <Ref> there is a minimal length factorization of u^-1v that starts with rr'.
	Then r' ≠ r.
	In addition we have r' ∈ [1, u^-1v], so r' ≻ r by ≺-minimality of r.
	
	The proof for the colexicographic order is similar.




After these preparatory lemmas, we are ready to prove the key results that lead to shellability.


	Fix an axial ordering ≺ of R_0, and let u ∈ [1,w] be an elliptic element.
	The interval [1,u] has at most one increasing maximal chain.

	Consider an irreducible horizontal component W_i ⊆ W, with associated hyperplane arrangement _i and horizontal prism C_i (with the notation of <Ref>).
	Let L be the intersection of all the hyperplanes of _i containing (u).
	Since (u) contains at least one axial vertex by <Ref>, only hyperplanes containing an axial vertex can occur, and so L is a flat in the subarrangement of _i consisting of the hyperplanes that contain a minimal face of C_i.
	Then L itself contains at least one minimal face F of C_i.
	By construction, the fixed set of any element of [1,u] ∩ W_i contains F (because it contains (u) and is a flat of _i).
	By <Ref>, this minimal face F is the fixed set of a maximal element u_i of [1,w] ∩ W_i.
	By <Ref>, [1,u] ∩ W_i ⊆ [1,u_i].
	
	The construction of the previous paragraph yields elements u_1, …, u_k, one for each irreducible horizontal component.
	Fix a point ∈ C_0 ∩ℓ.
	For i ∈{1,…, k}, apply <Ref> to the maximal element u_i and to the point ∈ℓ, and get an oriented line ℓ_i' with unit direction μ_i ∈(Φ_i).
	As usual, denote by μ the shortest vector in (w), which indicates the positive direction of the Coxeter axis ℓ.
	For ϵ > 0, consider the oriented line ℓ' = { + θμ' |θ∈} with basepoint a and direction μ' = μ + ϵμ_1 + … + ϵμ_k.
	By construction, its projection on the affine subspace through a parallel to (Φ_i) is precisely the line ℓ_i'. Therefore ℓ' and ℓ_i' intersect the hyperplanes of the reflections in W_u_i in the same order.
	
	Perturb the line ℓ' slightly, so that it becomes generic with respect to the hyperplanes of the reflections in W_u, and the basepoint  remains in C_0.
	If ϵ > 0 is small enough and the perturbation is small enough, the total ordering ≺_ℓ' of the finite set of reflections R_0 ∩ W_u has the following form:
	
  *  first, there are the positive vertical reflections of W_u (i.e. those that fix a point of ℓ above C_0), and r comes before r' if (r) ∩ℓ is below (r') ∩ℓ;
		
  *  then there are the horizontal reflections of W_u, and in each irreducible component W_i they are ordered as in ≺_ℓ_i';
		
  *  finally there are the negative vertical reflections of W_u, and again r comes before r' if (r) ∩ℓ is below (r') ∩ℓ.
	
	Notice that the relative order of the vertical reflections that fix the same point of ℓ can be different in ≺_ℓ' and in the axial ordering ≺.
	
	Let Φ_u ⊆Φ be the root system of W_u.
	By <Ref>, the total ordering ≺_ℓ' is a reflection ordering for W_u, with respect to the positive system Φ_u^+ ⊆Φ_u consisting of the roots that point towards the halfspaces containing the chamber C_0.
	We want to show that ≺_ℓ' is compatible with the Coxeter element u.
	For this, let Φ' ⊆Φ_u be an irreducible rank 2 induced subsystem.
	Let α and β be the simple roots of Φ' with respect to Φ' ∩Φ_u^+, with corresponding reflections r_α, r_β∈ W_u, and assume that r_α r_β∈ [1,u].
	We need to prove that r_α≺_ℓ' r_β.
	
  *  Case 1: r_α is vertical.
		By <Ref> (for the case Ã_n) and <Ref> (for the other cases), the right complement v of r_α is a Coxeter element for the finite parabolic subgroup W_v of W that fixes $̱, where$̱ is the unique vertex not fixed by r_α among the vertices of the axial chamber C immediately below (r_α) ∩ℓ.
		Since r_α r_β∈ [1,u] ⊆ [1,w], we have that r_β≤ v, so r_β fixes $̱ by <Ref>.
		
		 Let'be the reflection arrangement associated with the dihedral group r_α, r_β⊆ W_ugenerated byr_αandr_β(so its root system isΦ').
		Denote byC'the chamber of'containingC_0, i.e. the chamber corresponding to the positive systemΦ' ∩Φ_u^+.
		Denote byC”the chamber of'containingC.
		BothC'andC”haveH_α = (r_α)andH_β= (r_β)as their walls: this is true forC'by definition ofαandβ;H_αis a wall ofC”because it is a wall ofC;H_βis a wall ofC”because it containsb, which is a vertex ofCnot contained inH_α.
		So eitherC'andC”are the same chamber, or they are opposite chambers in'(becauseH_αandH_βare not orthogonal, so the arrangement'contains at least another hyperplane).
		SinceC_0andCare axial chambers, the Coxeter axisℓintersects bothC'andC”.
		The first hyperplane of'that intersectsℓaboveCisH_α, so the first hyperplane of'that intersectsℓ'aboveC”isH_α.
		
		 IfC' = C”, then the first hyperplane of'that intersectsℓ'aboveisH_α, and thereforer_α≺_ℓ' r_β.
		Suppose now thatC'andC”are opposite chambers in'.
		Sinceℓintersects bothC'andC”, andH_βseparatesC'andC”, we have thatr_βis also vertical.
		In addition,C'andC”are the two chambers of'that intersectℓ(resp.ℓ') in an unbounded subset ofℓ(resp.ℓ').
		Therefore, moving the basepointofℓ'fromℓ' ∩ C'toℓ' ∩ C”does not alter the total ordering≺_ℓ'of the reflections in r_α, r_β.
		As in the caseC' = C”, we conclude thatr_α≺_ℓ' r_β.
		
		
  *  Case 2:r_βis vertical.
		The argument is the same as for case 1, with the roles ofr_αandr_βexchanged.
		In this casevis the left complement ofr_β, andCis the axial chamber immediately above(r_β) ∩ℓ.
		
		
  *  Case 3: bothr_αandr_βare horizontal.
		Sinceαandβare not orthogonal, they must belong to the same irreducible horizontal componentΦ_i.
		Thenr_α r_β∈ [1,u] ∩ W_i ⊆ [1,u_i].
		By <Ref>, the reflection ordering forW_u_iinduced byℓ_i'is compatible withu_i, so we haver_α≺_ℓ_i' r_β, and thereforer_α≺_ℓ' r_β.
	

	We proved that≺_ℓ'is a reflection ordering forW_uwhich is compatible withu.
	By <Ref>, it makes the natural labelingλof[1,u]an EL-labeling.
	
	Suppose to have a≺-increasing maximal chain, which corresponds to a minimal length factorizationu = r_1r_2… r_mwithr_1 ≺ r_2 ≺…≺ r_m.
	Our aim is to show that the reflectionsr_1, r_2, …, r_mare uniquely determined.
	
	By definition of≺, the sequencer_1 ≺ r_2 ≺…≺ r_mconsists of an initial segmentr_1 ≺…≺ r_jof positive vertical reflections, a middle segmentr_j+1≺…≺ r_j'of horizontal reflections, and a final segmentr_j'+1≺…≺ r_mof negative vertical reflections.
	
	Reorder the reflections of the initial segment so that they are≺_ℓ'-increasing:r_σ(1)≺_ℓ' r_σ(2)≺_ℓ'…≺_ℓ' r_σ(j), for some permutationσof{1, …, j}.
	The relative order of vertical reflections that fix different points ofℓis the same in≺and≺_ℓ'(by construction ofℓ').
	By <Ref>, this means that the relative order of non-commuting vertical reflections is the same in≺and≺_ℓ'.
	Thereforer_σ(1) r_σ(2)… r_σ(j) = r_1 r_2 … r_j.
	Similarly, ifτis the permutation of{j'+1, …, m}such thatr_τ(j'+1)≺_ℓ' r_τ(j'+2)≺_ℓ'…≺_ℓ' r_τ(m), we haver_τ(j'+1) r_τ(j'+2)… r_τ(m) = r_j'+1 r_j'+2… r_m.
	
	Leth = r_j+1 r_j+2… r_j'∈ [1,u]be the product of the reflections in the middle segment.
	Since the total ordering≺_ℓ'makesλan EL-labeling of[1,u], the interval[1,h]has a unique≺_ℓ'-increasing maximal chain.
	It corresponds to a minimal length factorizationh = r_j+1' r_j+2' … r_j''withr_j+1' ≺_ℓ' r_j+2' ≺_ℓ'…≺_ℓ' r_j''.
	Sincehis horizontal, the reflectionsr_j+1', …, r_j''are horizontal.
	Therefore they come after the positive vertical reflections and before the negative vertical reflections, in the total ordering≺_ℓ'ofR_0 ∩ W_u.
	Putting everything together, we get
	L
		r_σ(1)≺_ℓ' r_σ(2)≺_ℓ'…≺_ℓ' r_σ(j)
≺_ℓ' r_j+1' ≺_ℓ' r_j+2' ≺_ℓ'…≺_ℓ' r_j'' 
≺_ℓ' r_τ(j'+1)≺_ℓ' r_τ(j'+2)≺_ℓ'…≺_ℓ' r_τ(m).
	
	The product of these reflections is equal tou.
	Since≺_ℓ'makesλan EL-labeling ofu, this factorization ofuis uniquely determined.
	In particular, the sets{ r_σ(1), …, r_σ(j)} = { r_1, …, r_j }and{ r_τ(j'+1), …, r_τ(m)} = { r_j'+1, …, r_m }are uniquely determined, and also the horizontal elementh = r_j+1' … r_j''is uniquely determined.
	Sincer_1 ≺ r_2 ≺…≺ r_j, andr_j'+1≺ r_j'+2≺…≺ r_m, the reflectionsr_1, …, r_jandr_j'+1, …, r_mare uniquely determined.
	Finally, the total order≺coincides with≺_on the horizontal reflections, and≺_makesλan EL-labeling of[1,h]by <Ref>.
	Then the reflectionsr_j+1, …, r_j'are uniquely determined, because they satisfyr_j+1≺_ r_j+2≺_…≺_ r_j'.

	Fix an axial ordering ≺ of R_0, and let u ∈ [1,w] be an hyperbolic element such that the Coxeter subgroup W_u ⊆ W is irreducible.
	The interval [1,u] has at most one increasing maximal chain.

	Suppose to have an increasing maximal chain, which corresponds to a minimal length factorization u = r_1r_2… r_m with r_1 ≺ r_2 ≺…≺ r_m.
	In this factorization, the horizontal reflections appear in a contiguous (possibly empty) middle segment.
	Since u is vertical, at least one of r_1 and r_m is a vertical reflection.
	Denote by _u the hyperplane arrangement associated with the irreducible affine Coxeter subgroup W_u ⊆ W.
	
	
  *  Case 1: r_1 is the first reflection of R_0 ∩ [1,u] with respect to the total order ≺.
		By <Ref>, r_1 is a positive vertical reflection.
		Then v = r_2… r_m is an elliptic element, and by <Ref> the reflections r_2, …, r_m are uniquely determined.
		More precisely, by <Ref>, the factorization v = r_2 … r_m has to be the unique colexicographically largest minimal length factorization of v.
		
		
  *  Case 2: r_1 is some other vertical reflection.
		Let C' be the axial chamber immediately below (r_1) ∩ℓ, and let C be the chamber of _u containing C'.
		By <Ref> there is a minimal length factorization of u that starts with r_1 and uses all the reflections with respect to the walls of C.
		Since r_1 is not the first reflection of R_0 ∩ [1,u], there is a reflection r with respect to a wall of C such that r ≺ r_1.
		Then r ≤ r_2… r_m.
		By <Ref>, the reflections r_2, …, r_m are uniquely determined, and by <Ref> the factorization r_2… r_m has to be the unique lexicographically smallest factorization of r_1 u.
		Therefore r_2 ≼ r ≺ r_1, which is impossible because r_1 ≺ r_2.
		
		
  *  Case 3: r_m is the last reflection of R_0 ∩ [1,u] with respect to the total order ≺.
		As in case 1, the reflections r_1, …, r_m-1 are uniquely determined.
		
		
  *  Case 4: r_m is some other vertical reflection.
		As in case 2, this is impossible.
	

	We have shown that there are at most two minimal length factorizations u = r_1 r_2 … r_m with r_1 ≺ r_2 ≺…≺ r_m: one where r_1 is the first reflection of R_0 ∩ [1,u], and r_2, …, r_m are uniquely determined (case 1), and one where r_m is the last reflection of R_0 ∩ [1,u], and r_1, …, r_m-1 are uniquely determined (case 3).
	By <Ref>, applied to the chamber of _u containing C_0, there is a minimal length factorization of u that starts with the first reflection of R_0 ∩ [1,u] and ends with the last reflection of R_0 ∩ [1,u].
	This means that, in case 1 (where r_1 is the first reflection of R_0 ∩ [1,u], and v = r_2… r_m), the interval [1,v] contains the last reflection of R_0 ∩ [1,u].
	Since r_2… r_m is the unique colexicographically largest factorization of v, the last factor r_m is the last reflection of R_0 ∩ [1,u].
	Then the factorization of case 1 coincides with the factorization of case 3.


Finally, we prove that axial orderings make the natural edge labeling of[1,w]an EL-labeling.


	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	Let λ([1,w]) → R_0 be the natural edge labeling of [1,w], where R_0 is totally ordered by an axial ordering.
	Every interval [u,v] in [1,w] has a unique increasing maximal chain, and this chain is both the lexicographically smallest and the colexicographically largest maximal chain of [u,v].
	In particular, λ is an EL-labeling of [1,w].

	By <Ref>, it is enough to consider intervals of the form [1,u], with u ∈ [1,w].
	In view of <Ref>, it only remains to show that [1,u] has at most one increasing maximal chain.
	If u is elliptic, this is done in <Ref>.
	If u is hyperbolic and the Coxeter subgroup W_u ⊆ W is irreducible, this is done in <Ref>.
	Suppose now that u is any hyperbolic element, and let u = u'h be its hyperbolic-horizontal decomposition (see <Ref>).
	Then u' is hyperbolic, the Coxeter subgroup W_u'⊆ W is irreducible, h is horizontal elliptic, and [1,u] = [1,u'] × [1,h].
	We have already proved that λ is an EL-labeling of [1,u'] and [1,h], so it is an EL-labeling of [1,u] by <Ref>.
	In particular, [1,u] has at most one increasing maximal chain.


§ DUAL CW MODELS FOR THE ORBIT CONFIGURATION SPACES


In this section, we introduce new finite CW models for the orbit configuration spaceY_Wof a Coxeter groupW.
Each of them is naturally included in the interval complex of one of the noncrossing partition posets associated withW.
We define them for any Coxeter group, not necessarily finite or affine.





	Let W be a Coxeter group.
	The reflection length of every Coxeter element is equal to the size of a set of simple reflections of W.

	Let w = s_1s_2… s_n be a Coxeter element, where S = {s_1, s_2, …, s_n} is a set of simple reflections.
	By the deletion condition <cit.> and the fact that S is a minimal generating set for W<cit.>, we obtain that s_1s_2… s_n is a reduced expression for w (meaning that w cannot be written as a product of less than n element of S).
	By <cit.>, the reflection length of w is the smallest natural number p such that s_i_1s_i_2… s_i_n-p = 1 for some choice of the indices 1 ≤ i_1 < i_2 < … < i_n-p≤ n.
	If p < n, then the relation s_i_1s_i_2… s_i_n-p = 1 allows to write a simple reflection as a product of other simple reflections, which is impossible because S is a minimal generating set.
	Therefore p=n.
<Ref> was proved in <cit.> for finite Coxeter groups, and in <cit.> for affine Coxeter groups.

LetWbe a Coxeter group, andRits set of reflections.
Fix a set of simple reflectionsS = {s_1, s_2, …, s_n }⊆ R, and a Coxeter elementw = s_1s_2 … s_n.
Denote byK_Wthe interval complex associated with the noncrossing partition poset[1,w]^W.
LetX_Wbe the Salvetti complex ofW, and recall from <Ref> that its cells are indexed by the simplicial complex

    Δ_W = { T ⊆ S |the standard parabolic subgroup W_T is finite}.


For everyT ∈Δ_W, denote byw_Tthe product of the elements ofTin the same relative order as in the lists_1,s_2,…,s_n.
Thenw_Tis a Coxeter element of the parabolic subgroupW_T, and it belongs to[1,w]^Wby <Ref> and <Ref>.





	For every T ⊆ S we have [1,w_T]^W_T = [1,w_T]^W, and the length functions of W_T and W agree on these intervals.


	
	By <cit.>, the length function of the parabolic subgroup W_T agrees with the length function of W.
	By <Ref>, l(w_T) = |T|.
	
	By <cit.> (see also <cit.>), the Hurwitz action is transitive on the minimal length factorizations of w_T as a product of reflections of W.
	There is at least one minimal length factorization of w_T that uses only reflections of W_T, and therefore this is true for all minimal length factorizations.

	This means that the interval [1,w_T] is the same in W_T (using the reflections of W_T as the generating set) and in W.


Thanks to the previous lemma, for everyT ⊆ Swe can safely write[1,w_T]in place of[1,w_T]^W_T = [1,w_T]^W, without the need to specify the ambient group.


	Let X_W' be the finite subcomplex of K_W consisting of the simplices [x_1|x_2|…|x_d] ∈ K_W such that x_1x_2… x_d ∈ [1,w_T] for some T ∈Δ_W.
	
	If W is finite, then S ∈Δ_W and therefore X_W' = K_W.
	In this case, the interval complex K_W is a classifying space for the dual Artin group W_w (by <Ref>), which is naturally isomorphic to the Artin group G_W (by <Ref>).



For everyT ∈Δ_W, the complexX_W'has a subcomplex consisting of the simplices[x_1|x_2|…|x_d]such thatx_1x_2… x_d ∈ [1,w_T] = [1,w_T]^W_T.
This is exactly the interval complex associated with[1,w_T]^W_T, which coincides withX_W_T'and is a classifying space for the Artin groupG_W_Tby <Ref>.

By definition,X_W'is the union of all subcomplexesX_W_T'forT ∈Δ_W.
Similarly, the Salvetti complexX_Wis the union of the Salvetti complexesX_W_TforT ∈Δ_W.
EachX_W_Tis a classifying space forG_W_T, because theK(π, 1)conjecture holds for spherical Artin groups <cit.>.



	For every Coxeter group W, the complex X_W' is homotopy equivalent to the Salvetti complex X_W and to the orbit configuration space Y_W.
	

	
	
	
	
	
	
	
	
	
	
	
	Since X_W is homotopy equivalent to Y_W, it is enough to show that X_W' is homotopy equivalent to X_W.
	To keep our notation uncluttered, throughout this proof we indicate X_W, X_W', X_W_T, X_W_T' by X, X', X_T, X_T', respectively.
	
	For every T ∈Δ_W, both complexes X_T and X_T' are classifying spaces for the Artin group G_W_T.

	We are going to inductively construct homotopy equivalences φ_T  X_T→ X'_T satisfying the following naturality property: for all Q ⊆ T ∈Δ_W, there is a commutative diagram
	

			X_Q[r, "φ_Q"] [d, hook]
			    X'_Q[d, hook] 

			X_T[r, "φ_T"]
			    X'_T.
		

	The construction of φ_T is the following, assuming to have already constructed φ_Q for all Q ⊊ T.
	
  *  If T = ∅, then X_T and X'_T are single points, and there is only one map φ_T between them.
		
		
  *  If |T| = 1, then both X_T and X'_T consist of one oriented 1-cell attached to one 0-cell.
		Define φ_T as any orientation-preserving cellular homeomorphism.
		
		
  *  Let |T| = 2, say T = {s, s'}.
		We can extend φ_{s}∪φ_{s'} X_{s}∪ X_{s'}→ X_T' to the 2-cell of X_T, and obtain a map φ_T X_T → X_T' such that the induced map (φ_T)_* π_1(X_T, X_∅) →π_1(X_T', X_∅') is an isomorphism (see the proof of <cit.>).
		Since X_T and X_T' are classifying spaces, we have that φ_T is a homotopy equivalence.
		
		
  *  Let |T| ≥ 3. By construction, the map ⋃_Q⊊ Tφ_Q ⋃_Q⊊ T X_Q → X_T' induces an isomorphism on the fundamental groups (which are both isomorphic to the Artin group G_W_T).
		Extend this map to the |T|-cell of X_T, to get a map φ_T  X_T → X_T' which also induces an isomorphism on the fundamental groups (as in the proof of <cit.>).
		As before, φ_T is a homotopy equivalence.
	
	Gluing together all these maps, we obtain a map φ X_W → X'_W.
	This is a homotopy equivalence by a repeated application of the gluing theorem for adjunction spaces <cit.>.

	For every Coxeter group W, the fundamental group of X_W' is isomorphic to the Artin group G_W.

	The complex X_W' depends on the Coxeter element w and on the set of simple reflections S.
	However, since all Coxeter elements of a finite parabolic subgroup W_T are geometrically equivalent, the f-vector of X_W' depends only on W (it can be computed via inclusion-exclusion in terms of the subcomplexes K_W_T).



In view of <Ref>, theK(π,1)conjecture holds for an Artin groupG_Wif and only ifX_W'is a classifying space.
The following is an alternative characterization of the cells of the complexX_W'in the affine case.


	Let W be an irreducible affine Coxeter group, with a set S of simple reflections and a Coxeter element w obtained as a product of the elements of S.
	Denote by C_0 the chamber of the Coxeter complex associated with S.
	A simplex [x_1|x_2|…|x_d] ∈ K_W belongs to X_W' if and only if x_1x_2… x_d is an elliptic element that fixes at least one vertex of C_0.

	For every subset T ⊆ S with |T| = |S|-1, the fixed set of the parabolic Coxeter element w_T is given by one of the vertices of C_0 by <Ref>.
	Conversely, every vertex of C_0 is the fixed set of exactly one such parabolic Coxeter element.
	We conclude using <Ref>.
[Dual complexes for Ã_2 and G̃_2]
	The f-vector of X_W' is (1,9,9) if W is of type Ã_2, and (1,11,11) if W is of type G̃_2.
	For Ã_2, the complex X_W' is explicitly described in <Ref> and <Ref>.
	Notice that the Salvetti complex X_W is much smaller, as in both cases its f-vector is (1,3,3).


§ CLASSIFYING SPACES FOR DUAL AFFINE ARTIN GROUPS


LetWbe an irreducible affine Coxeter group, andwone of its Coxeter elements.
In this section, we prove that the interval complexK_Wassociated with[1,w]^Wis a classifying space for the dual Artin groupW_w.
This is somewhat surprising since the interval[1,w]^Wis not a lattice in general.

As usual, letWact by Euclidean isometries on^n, wherenis the rank ofW.
LetR ⊆ Wbe the set of reflections, and denote byR_andR_$̌ the horizontal and vertical reflections of [1,w]^W, respectively.
To proceed, we briefly recall the construction of McCammond and Sulway that leads to braided crystallographic groups <cit.>.
For this, new groups of Euclidean isometries are introduced.


  *  The diagonal groupD, generated by R_ and T.
	Here T is the (finite) set of all translations of [1,w]^W.
	Translations are assigned a weight of 2.
	
  *  The factorable groupF, generated by R_ and by a set T_F of factored translations.
	There are k factored translations t_1, …, t_k for each translation t ∈ T, and they satisfy t_1… t_k = t, where k is the number of irreducible components of the horizontal root system Φ_.
	Factored translations are assigned a weight of 2/k.
	
  *  The crystallographic groupC, generated by R_, R_$̌, andT_F.

The diagonal groupDis included in bothWandF, and all of them are included in the crystallographic groupC.
By <cit.>, the associated intervals are related as follows:

    [1,w]^C    = [1,w]^W ∪ [1,w]^F 
    
    [1,w]^D    = [1,w]^W ∩ [1,w]^F.


The intervals[1,w]^Dand[1,w]^Fare finite, whereas[1,w]^Wand[1,w]^Care infinite.
The factored translations are introduced so that the intervals[1,w]^Fand[1,w]^Care balanced lattices <cit.>.
On the other hand, the intervals[1,w]^Dand[1,w]^Ware lattices if and only if the horizontal root systemΦ_is irreducible (i.e.k=1), in which caseD=FandW=C.

From these new intervals, one can construct the interval groupsD_w,F_w, andC_w. The groupC_wis called the braided crystallographic group. 
The inclusions between the four intervals induce inclusions between the corresponding interval groups:D_w ↪ W_w,D_w ↪ F_w,W_w ↪ C_w, andF_w ↪ C_w<cit.>.
Since the intervals[1,w]^Fand[1,w]^Care lattices, the interval groupsF_wandC_ware Garside groups (<Ref>) and the corresponding interval complexesK_FandK_Care classifying spaces (<Ref>).
A consequence of the relations between the four intervals is thatK_C = K_W ∪ K_FandK_D = K_W ∩ K_F, whereK_Dis the interval complex associated with[1,w]^D.
As noted in <cit.>, the cover ofK_Ccorresponding to the subgroupW_w ⊆ C_wis a finite-dimensional classifying space for the (dual) Artin groupW_w.
However this cover is not very explicit, and it is difficult to work with it in practice.


LetHbe the subgroup ofDgenerated byR_.
With the notation of <Ref> we haveH = W_1 ×…× W_k, whereW_iis the subgroup generated by the reflections associated with thei-th irreducible componentΦ_iof the horizontal root systemΦ_.
Recall thatW_iis a Coxeter group of typeÃ_n_i, wheren_iis the rank ofΦ_i.
By <cit.>, the horizontal part[1,w]^W ∩ Hof the interval[1,w]^Wdecomposes as

    [1,w]^W ∩ H = ( [1,w]^W ∩ W_1 ) ×…×( [1,w]^W ∩ W_k ),

and the single factors are described in <Ref> (a more direct proof of this decomposition can be derived in a similar way to the proof of <Ref>).
LetH_wbe the group with generating setR_and subject only to the relations visible in[1,w]^W ∩ H.
It is a subgroup ofD_w<cit.>, and it decomposes as a direct product ofkArtin groups of typesÃ_n_1, …, Ã_n_k, one for each irreducible component.



















LetK_Hbe the subcomplex ofK_Dconsisting of the simplices[x_1|x_2|…|x_d]such thatx_1x_2… x_d ∈ H.
The fundamental group ofK_His naturally isomorphic toH_w.
Denote byK_ithe subcomplex ofK_Hconsisting of the simplices[x_1|x_2|…|x_d]such thatx_1x_2… x_d ∈ [1,w]^W ∩ W_i.
The fundamental group ofK_iis an Artin group of typeÃ_n_i.


K_H is homeomorphic to K_1×…× K_k.
	
	We claim that K_H is a triangulation of the natural cellular structure of K_1×…× K_k.
	To show this, we explicitly construct a homeomorphism ψ K_1×…× K_k→ K_H.
	Consider a cell of K_1×…× K_k, which is a product of simplices
	
    [x_11| x_12| … | x_1d_1] ×…× [x_k1| x_k2| … | x_kd_k].

	This is realized as Δ^d_1×…×Δ^d_k⊆^d_1+…+d_k.
	Consider a point p ∈Δ^d_1×…×Δ^d_k, with coordinates given by
	
    (a_11,…, a_1d_1,…,a_k1,…,a_kd_k) ∈^d_1+…+d_k.

	Assume for now that the coordinates of p are pairwise distinct.
	Then there is a unique enumeration γ{1,…,d_1+…+d_k}→{11, …, 1d_1, …, k1, …, kd_k } of the indices such that a_γ(1)≥ a_γ(2)≥…≥ a_γ(d_1+…+d_k).
	Notice that, in this enumeration, the relative order of indices in the same horizontal component is preserved.
	Define ψ(p) as the point of [x_γ(1)| x_γ(2) |… | x_γ(d_1+…+d_k)] with coordinates
	
    (a_γ(1), a_γ(2), …, a_γ(d_1+…+d_k)) ∈Δ^d_1+…+d_k.

	If the coordinates of p are not pairwise distinct, there are multiple choices for the enumeration γ, and any choice gives the same definition of ψ(p).
	See <Ref> for some examples.
	Using the decomposition (<ref>), we obtain that ψ is a homeomorphism.
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Denote byφ [1,w]^W → [1,w]^Wthe conjugation byw:φ(u) = w^-1 u w.

K_D is homeomorphic to K_H × [0,1] / ∼, where the relation ∼ identifies [x_1|x_2|…|x_d] ×{1} and [φ(x_1) | φ(x_2) | … | φ(x_d)] ×{0} for every simplex [x_1|x_2|…|x_d] of K_H.
	
	Let Z = K_H × [0,1]   / ∼.
	Similarly to <Ref>, we show that K_D is a triangulation of the natural cell structure of Z, by explicitly constructing a homeomorphism ψ Z → K_D.
	Notice that K_H ×{0}⊆ Z is naturally included in K_D: we define ψ|_K_H ×{0} as the natural homeomorphism with K_H ⊆ K_D.
	Consider now a cell of K_H × [0,1] of the form
	
    [x_1|x_2|…|x_d] × [0,1].

	This cell is realized as Δ^d × [0,1] ⊆^d+1.
	Then a point p in this cell has coordinates (a_1,a_2,…,a_d,t), with 1 ≥ a_1 ≥…≥ a_d ≥ 0 and t ∈ [0,1].
	Let y be the right complement of x_1x_2… x_d, so that x_1x_2… x_d y = w.
	Notice that y is hyperbolic, so in particular y ≠ 1.
	Assume for now that none of the coordinates a_1,…, a_d is equal to 1-t.
	Then there is a unique index i ∈{0,…,d} such that a_1 ≥…≥ a_i ≥ 1-t ≥ a_i+1≥…≥ a_d.
	Let σ = [x_i+1|… |x_d|y|φ(x_1)|…|φ(x_i)].
	Define ψ(p) as the point of σ with coordinates (t+a_i+1,…,t+a_d,t,t+a_1-1,…,t+a_i-1) ∈Δ^d+1.
	If some of the coordinates a_1,…, a_d are equal to 1-t, choose any index i as above. Different choices of i give the same point of K_D.
	See <Ref> for some examples.
	
	The definition of ψ on different cells is coherent.
	We only explicitly check that the definition on [x_1|x_2|…|x_d] × [0,1] agrees with the definition on [x_1|x_2|…|x_d] ×{1}, since this is where the non-trivial gluing occurs.
	Consider a point p ∈ [x_1|x_2|…|x_d] ×{1}, with coordinates (a_1,…,a_d,1) ∈Δ^d × [0,1].
	
  *  Since [x_1|…|x_d]×{1} is identified with [φ(x_1)|…|φ(x_d)] ×{0}, we have that ψ(p) is the point of [φ(x_1)|…|φ(x_d)] with coordinates (a_1,…,a_d).
		
		
  *  As an element of [x_1|…|x_d] × [0,1], the same point p is sent to the point of [y|φ(x_1)|…|φ(x_d)] with coordinates (1,a_1,…,a_d).
		By definition of the faces in an interval complex (<Ref>), this point is the same as the point of [φ(x_1)|…|φ(x_d)] with coordinates (a_1,…,a_d).
	
	Therefore the two definitions of ψ agree in this case.
	
	Every maximal cell of K_D is of the form [x_1|…|x_i|y|x_i+1'|…|x_d'] where y is hyperbolic and each x_j and x_j' is horizontal elliptic.
	Thus ψ is a homeomorphism.



An immediate consequence of <Ref> is thatD_w = ⋉ H_w, whereis the cyclic subgroup ofD_wgenerated byw.


K_i is a classifying space for the affine Artin group of type Ã_n_i.
	
	By <Ref>, we can assume until the end of this proof that H = W_i is the unique irreducible horizontal component of W.
	Therefore [1,w]^D = [1,w]^F is a lattice, D_w is a Garside group, and K_D is a classifying space for D_w.
	By <Ref>, there is a covering map
	
    ρ K_H ×→ K_D

	which corresponds to the subgroup H_w of D_w.
	Then K_H × is a classifying space.
	Since K_H ×≃ K_H = K_i, also K_i is a classifying space.

	The complex K_i is closely related to the complexes X_W_i' introduced in <Ref>.
	Indeed, K_i is obtained by gluing the interval complexes associated with the noncrossing partition lattices of type A_n_i corresponding to the maximal proper standard parabolic subgroups of W_i.
	However, the Coxeter elements of these parabolic subgroups are not below a common Coxeter element of Ã_n_i (see also <Ref>).
K_D is a classifying space for D_w.
	
	By <Ref>, we have that K_H ≅ K_1×…× K_k.
	Each factor is a classifying space by <Ref>, therefore K_H is a classifying space for H_w.
	As discussed in the proof of <Ref>, there is a covering map ρ K_H ×→ K_D by <Ref>.
	Therefore K_D is a classifying space.



We can finally prove thatK_Wis a classifying space.


	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	The interval complex K_W is a classifying space for the dual Artin group W_w.
	
	Consider the universal cover ρK̃_C → K_C of the interval complex K_C.
	Recall that K_D = K_W ∩ K_F and K_C = K_W ∪ K_F, and therefore ρ^-1(K_D) = ρ^-1(K_W) ∩ρ^-1(K_F) and K̃_C = ρ^-1(K_C) = ρ^-1(K_W) ∪ρ^-1(K_F).
	Then there is a Mayer-Vietoris long exact sequence
	
    …→ H_i(ρ^-1(K_D)) → H_i(ρ^-1(K_W)) ⊕ H_i(ρ^-1(K_F)) → H_i(K̃_C) →…

	where all homology groups are with integer coefficients.
	Since D_w = π_1(K_D) is a subgroup of C_w = π_1(K_C), we have that ρ^-1(K_D) is a union of (infinitely many) disjoint copies of the universal cover K̃_D of K_D.
	Similarly, ρ^-1(K_W) is a union of disjoint copies of the universal cover K̃_W of K_W, and ρ^-1(K_F) is a union of disjoint copies of the universal cover K̃_F of K_F.
	Since K_F and K_C are classifying spaces, both K̃_F and K̃_C are contractible.
	By <Ref>, K̃_D is also contractible.
	Then, for i ≥ 1, the homology groups H_i(ρ^-1(K_D)), H_i(ρ^-1(K_F)), and H_i(K̃_C) vanish.
	By the Mayer-Vietoris long exact sequence, H_i(ρ^-1(K_W)) also vanishes for i ≥ 1.
	This means that K̃_W has a trivial reduced homology, so it is contractible by a standard application of the Whitehead and Hurewicz theorems (see <cit.>).


§ FINITE CLASSIFYING SPACES



LetWbe an irreducible affine Coxeter group, with a fixed Coxeter elementw.
In this section we show that the interval complexesK_WandK_Cdeformation retract onto finite subcomplexesK_W' ⊆ K_WandK_C' ⊆ K_C.
In the case ofK_W, this is an intermediate step to prove theK(π,1)conjecture, whereas forK_Cthis proves that the braided crystallographic groupC_whas a classifying spaceK_C'with a finite number of cells.
The notation is the same as in <Ref>.

As noted in the proof of <cit.>, there is no minimal length factorization ofwinCthat includes both a factored translation and a vertical reflection.
Recall that every element of[1,w]^C ∖ [1,w]^Wis hyperbolic.



	Let σ = [x_1|x_2|…|x_d] be a d-simplex of K_C, with d ≥ 1.
	Then exactly one of the following occurs:
	
  *  every x_i is elliptic, and at least one is vertical;
		
  *  every x_i is horizontal elliptic or hyperbolic.
	

	We shall divide the proof into four cases.
	
  *  If at least one x_i is not in [1,w]^W, then no minimal length factorization of any x_i includes vertical reflections. In particular, no x_i is vertical elliptic, so (ii) holds and (i) does not.
		In the remaining cases, assume that x_i ∈ [1,w]^W for all i.
		
		
  *  If every x_i is horizontal elliptic, then (ii) holds and (i) does not.
		
		
  *  Suppose that x_j is hyperbolic for some index j. Then x_1… x_j is also hyperbolic, and therefore its right complement y is horizontal elliptic. Every x_i for i>j is below y in [1,w]^W, and so is horizontal elliptic.
		By a similar argument, every x_i for i<j is horizontal elliptic.
		Then (ii) holds and (i) does not.
		
		
  *  If there is at least one vertical elliptic element and there are no hyperbolic elements, (i) holds and (ii) does not. 


Recall that(K_C)denotes the face poset ofK_C.
Consider the poset mapη(K_C) →defined by

    η([x_1|x_2|…|x_d]) =
    
    d    if x_1 x_2 … x_d = w
    
    d+1    otherwise.

We want to describe the connected components of a fiberη^-1(d)in the Hasse diagram of(K_C).
Letσ, τbe two simplices in the same fiberη^-1(d). We have thatτis a face ofσif and only ifσ = [x_1|x_2|…|x_d]withx_1 x_2 … x_d=wand eitherτ = [x_2|x_3|… |x_d]orτ = [x_1|x_2|…|x_d-1].
Therefore, a connected component ofη^-1(d)has the following form:

 (0) at (0,0) [x_1|…|x_d];
	
	 (1) at (1.5, ) [x_2|…|x_d];
	 (2) at (3, 0) [x_2|…|x_d+1];
	 (3) at (4.5, ) [x_3|…|x_d+1];
	 (4) at (6,0) [ ];
	 (3a) at ((3)!0.5!(4)) ;
	
	
	 (-1) at (-1.5, ) [x_1|…|x_d-1];
	 (-2) at (-3, 0) [x_0|…|x_d-1];
	 (-3) at (-4.5, ) [x_0|…|x_d-2];
	 (-4) at (-6, 0) [ ];
	 (-3a) at ((-3)!0.5!(-4)) ;
	
	 (0.south) -> (1.north);
	 (2.south) -> (1.north);
	 (2.south) -> (3.north);
	[dashed] (3.north) -> (3a);
	
	 (0.south) -> (-1.north);
	 (-2.south) -> (-1.north);
	 (-2.south) -> (-3.north);
	[dashed] (-3.north) -> (-3a);
	

wherex_i x_i+1… x_i+d-1 = wfor alli.
Define a d-fiber component (or simply fiber component) as a connected component of the fiberη^-1(d).
As described above, ad-fiber component has an associated sequence(x_i)_i∈of elements of[1,w]^Csuch that the product of anydconsecutive elements isw.
This sequence is well-defined up to a translation of the indices.


Letφ [1,w]^C → [1,w]^Cbe the conjugation by the Coxeter elementw:φ (u) = w^-1uw.
Notice that, if(x_i)_i∈is the sequence associated with ad-fiber component, we haveφ(x_i) = x_i+dfor alli∈.
Sinceφrestricts to a map[1,w]^W → [1,w]^W, every fiber component is either disjoint from(K_W)or contained in(K_W), where(K_W)is the face poset ofK_W.
The same is true for the subcomplexesK_DandK_F.


	Let u ∈ [1,w]^C. The set {φ^j(u) | j ∈} is infinite if and only if u is vertical elliptic.
	
	Suppose that {φ^j(u) | j ∈} is infinite.
	Since [1,w]^C has only a finite number of horizontal elliptic and hyperbolic elements, at least one element of {φ^j(u) | j ∈} is vertical elliptic. Then u is vertical elliptic.
	
	Conversely, let u be a vertical elliptic element, and suppose by contradiction that φ^j(u) = u for some j ∈∖{0}.
	Let w^p be a power of w that acts as a translation in the positive direction of the Coxeter axis ℓ, where p is a positive integer (see <Ref>).
	We have that φ^pj(u) = u, so u commutes with the (non-trivial) translation w^pj.
	By <cit.>, (u) is invariant under w^pj.
	Then (ℓ) ⊆((u)).
	We have that (u) = ((u)) because u is elliptic, and that ((u)) is orthogonal to ((u)) by <cit.>.
	Therefore u is horizontal, and this is a contradiction.

	Let σ∈(K_C).
	The fiber component containing σ is infinite if and only if σ is of type (i) in <Ref>.
	In particular:
	
  *  every infinite fiber component is contained in (K_W);
		
  * (K_F) is the union of all finite fiber components;
		
  * (K_D) is the union of all finite fiber components of (K_W).
	
	Let σ = [x_1|x_2|…|x_d].
	If σ is of type (i), at least one x_i is vertical elliptic. Then the set { x_i+jd = φ^j(x_i) | j ∈} is infinite by <Ref>, so the fiber component of σ is infinite.
	The simplices of type (i) are in (K_W), so the component is contained in (K_W).
	
	If σ is of type (ii), then every x_i is horizontal elliptic or hyperbolic, so σ∈(K_F).
	Since (K_F) is finite, the fiber component of σ is finite.
	Conversely, every simplex of (K_F) is of type (ii), and so its fiber component is finite.
	
	The very last point follows from the fact that K_D = K_W ∩ K_F.

	Let ⊆(K_W) be a finite d-fiber component.
	Then there exists a simplex [x_1|x_2|…|x_d-1] ∈ such that x_1x_2… x_d-1 is horizontal elliptic.
	
	Consider any d-simplex σ = [x_1|x_2|…|x_d] ∈, with x_1 x_2 … x_d = w.
	Since  is finite, at least one x_i is hyperbolic by <Ref>.
	Suppose without loss of generality that x_d is hyperbolic.
	Then its left complement x_1x_2… x_d-1 is horizontal elliptic (here we are using the fact that σ∈(K_W)).
	This completes the proof, because [x_1|x_2|…|x_d-1] ∈.

	Let ⊆(K_W) be an infinite d-fiber component.
	Then there exists a simplex [x_1|x_2|…|x_d-1] ∈ such that x_1x_2… x_d-1 is vertical elliptic.
	
	Consider any d-simplex [x_1|x_2|…|x_d] ∈, with x_1 x_2 … x_d = w.
	Since  is infinite, at least one x_i is vertical elliptic by <Ref>.
	Suppose without loss of generality that x_d is vertical elliptic.
	Then its left complement x_1x_2… x_d-1 is also vertical elliptic, and [x_1|x_2|…|x_d-1] ∈.



From now on, fix an axial chamberC_0of the Coxeter complex.
IfSis the set of simple reflections associated withC_0, the Coxeter elementwcan be written as the product of the elements ofSby <Ref> (for the caseÃ_n) and <Ref> (for the other cases).
LetX_W' ⊆ K_Wbe the complex introduced in <Ref>.
By <Ref>, it consists of the simplices[x_1|x_2|…|x_d]ofK_Wsuch thatx_1x_2… x_dfixes a vertex ofC_0.



	Let ⊆(K_W) be a d-fiber component.
	Then there exists a simplex [x_1|x_2|…|x_d-1] ∈ such that x_1 x_2 … x_d-1 is elliptic and fixes a vertex of C_0.
	In other words, ∩(X_W') ≠∅.
	
	By <Ref>, there exists a simplex σ = [x_1|x_2|…|x_d-1] ∈ such that x_1 x_2… x_d-1 is elliptic.
	By <Ref>, x_1 x_2… x_d-1 fixes an axial vertex.
	By <Ref> (for the case Ã_n) and <Ref> (for the other cases), every axial vertex can be written uniquely as w^j(b) for some vertex b of C_0.
	Then, up to a conjugation by a power of w (i.e. up to a translation of the indices in the sequence (x_i)_i∈), we can assume that x_1 x_2 … x_d-1 fixes a vertex of C_0.


	The face poset (K_C) of the interval complex K_C contains only a finite number of fiber components.

	Every fiber component contained in (K_W) intersects (X_W') by <Ref>.
	Since (X_W') is finite, (K_W) contains only a finite number of fiber components.
	If  is a fiber component not contained in (K_W), by <Ref> we have that  is finite and ⊆(K_F).
	Since (K_F) is finite, it contains only a finite number of fiber components.




We are finally able to show thatK_CandK_Wdeformation retract onto finite subcomplexesK_C' ⊆ K_CandK_W' ⊆ K_W, respectively.


	Let K be either K_C or K_W.
	A nice subcomplex of K is a subcomplex K' ⊆ K such that
	
  *  every finite fiber component ⊆(K) is also contained in (K');
		
  *  for every infinite fiber component ⊆(K), the intersection ∩(K') is non-empty and its Hasse diagram is connected.
	
	Let K be either K_C or K_W.
	
  * K deformation retracts onto every nice subcomplex K'.
		
  *  Finite nice subcomplexes of K exist.
	
	For part (a), on every infinite fiber component  consider the only acyclic matching _ with critical simplices given by ∩(K'). Existence and uniqueness of _ follow from the fact that ∩(K') is non-empty and its Hasse diagram is connected.
	By <Ref>, the union of the matchings _ is an acyclic matching with the desired set of critical simplices.
	This matching is also proper.
	We conclude using the main theorem of discrete Morse theory (<Ref>).
	
	For part (b), recall that (K) has only a finite number of (finite or infinite) fiber components by <Ref>.
	Then it is enough to inductively choose a finite non-empty interval in the Hasse diagram of every infinite d-fiber component , starting from d=(K) (the highest possible value of d) and going down to d=1, so that every simplex in the boundary of a chosen simplex is also chosen.


SinceK_Cis a classifying space forC_w, we immediately obtain the following.


	Let W be an irreducible affine Coxeter group, and w one of its Coxeter elements.
	The braided crystallographic group C_w admits a classifying space with a finite number of cells.
	


We end this section by noticing that there is a canonical choice of a nice subcomplexK'ofK(whereKis eitherK_WorK_C): for every infinite fiber component,K'contains all the simplices between the first and the last simplex belonging toX_W'; in addition,K'contains all the finite fiber components ofK.



	Let K be either K_C or K_W.
	Then K' is a nice subcomplex of K.
	
	First, we check that K' is a subcomplex.
	The finite fiber components of K form a subcomplex (K_D or K_F) by <Ref>.
	Let σ be a simplex of K' which belongs to an infinite fiber component .
	Then, in , the simplex σ is between two simplices σ_1,σ_2 ∈ X_W' (if σ∈ X_W', we have σ_1= σ_2= σ).
	Then a face τ of σ is either in the same fiber component , or is between two faces of σ_1 and σ_2, in the fiber component of τ. Therefore τ∈(K').
	
	By <Ref>, every infinite fiber component contains at least one simplex of X_W'.
	Then K' satisfies the conditions of <Ref>.


We call this subcomplexK'the canonical nice subcomplex ofK.
By construction, it containsX_W'as a subcomplex.
In general,K'is not the smallest nice subcomplex ofK.


[Fiber components of Ã_2]
	For W of type Ã_2, the fiber components of K_W = K_C are shown in <Ref>, using the notation of <Ref>.
	The first 2 fiber components are finite, and they form the subcomplex K_D = K_F.
	The other 7 fiber components are infinite.
	Black nodes correspond to simplices in (X_W'), and white nodes correspond to simplices in (K_W') ∖(X_W'), where K_W' is the canonical subcomplex of K_W.
	The shown simplices are exactly those in (K_W').
	

§ THE K(PI,1) CONJECTURE


In this section, we prove theK(π,1)conjecture for affine Artin groups.
It is enough to consider the irreducible case.

LetWbe an irreducible affine Coxeter group, andRits set of reflections.
Fix a Coxeter elementwand an axial chamberC_0of the Coxeter complex.
LetS ⊆ Rbe the set of simple reflections associated withC_0.
By <Ref> (for the caseÃ_n) and <Ref> (for the other cases), the Coxeter elementwcan be written as the product of the elements ofS, sayw = s_1s_2… s_n+1(wherenis the rank ofW).
Let{p_i}_i ∈be the sequence of points of the Coxeter axisℓthat are fixed by at least one vertical reflection ofW(see <Ref> and <Ref>).
Enumerate these points so thatp_0is belowC_0andp_1is aboveC_0.

LetK_Wbe the interval complex associated with the noncrossing partition poset[1,w] = [1,w]^W,K_W' ⊆ K_Wits canonical nice subcomplex (introduced at the end of <Ref>), andX_W' ⊆ K_W'the complex introduced in <Ref>.
By <Ref>,X_W'consists of the simplices[x_1|x_2|…|x_d]ofK_Wsuch thatx_1x_2… x_dfixes a vertex ofC_0.

Recall thatK_Wis a classifying space for the dual Artin groupW_w(<Ref>),K_Wdeformation retracts ontoK_W'(<Ref>), andX_W'is homotopy equivalent to the orbit configuration spaceY_W(<Ref>).
Then, in order to prove theK(π,1)conjecture for the Artin groupG_W = π_1(Y_W), it is enough to show thatK_W'deformation retracts ontoX_W'.
This also implies that the dual Artin groupW_wis isomorphic to the Artin groupG_W, thus giving a new proof of <cit.>.

To show thatK_W'deformation retracts ontoX_W', we are going to use discrete Morse theory.
Specifically, we need to construct an acyclic matching on the face poset(K_W')such that the set of critical simplices is exactly(X_W').

Denote byφ [1,w] → [1,w]the conjugation byw:φ(u) = w^-1uw.
Let≺be an axial ordering of the set of reflectionsR_0 = R ∩ [1,w](see <Ref>) that satisfies the following compatibility property: ifr,r' ∈ R_0fix the same point ofℓ, andr ≺ r', thenφ(r) ≺φ(r').
Although not strictly necessary, this compatibility property will make the proof of <Ref> simpler.

By <Ref>, every elementu ∈ [1,w]has a unique minimal length factorizationu = r_1 r_2 … r_mas a product of reflections such thatr_1 ≺ r_2 ≺…≺ r_m.
We call it the increasing factorization ofu.
<Ref> also implies thatr_1is the≺-smallest reflection ofR_0 ∩ [1,u], andr_mis the≺-largest.
As in <Ref>, we say that a vertical reflection is positive if it fixes a point ofℓaboveC_0, and negative otherwise.

Sincewacts on the Coxeter axisℓas a translation in the positive direction, we have that(φ(r)) ∩ℓis below(r) ∩ℓfor every vertical reflectionr ∈ R_0.
In addition, if(r) ∩ℓis above(r') ∩ℓ(for some vertical reflectionsr,r'), then(φ^j(r)) ∩ℓis above(φ^j(r')) ∩ℓfor allj ∈.



	For every vertical reflection r ∈ R_0, there exists a unique j ∈ such that φ^j(r) is one of the n+1≺-smallest reflections in R_0.
	In addition, j ≥ 0 if and only if r is positive.
	
	If w is a bipartite Coxeter element (see <Ref>), the n+1≺-smallest reflections are the ones that fix p_1 or p_2.
	Let p_i be the point of ℓ which is fixed by r.
	By <Ref>, there is a unique j such that w^-j(p_i) is p_1 or p_2, and specifically j = ⌊i-1/2⌋.
	We have j ≥ 0 if and only if r is positive.
	Then φ^j(r) fixes p_1 or p_2, so it is among the n+1≺-smallest reflections of R_0.
	
	Suppose now that w is a (p,q)-bigon Coxeter element (with p+q=n+1) in a Coxeter group W of type Ã_n.
	By part (i) of <Ref>, the n+1≺-smallest reflections are those that fix one of p_1, p_2, …, p_m with m = p+q/(p,q).
	By <Ref>, for every i ∈ there is a unique j ∈ such that w^-j(p_i) ∈{p_1, p_2, …, p_m}.
	We conclude as in the bipartite case.

	Let r ∈ R_0 be a vertical reflection.
	The right complement of r fixes a vertex of C_0 if and only if r is among the n+1≺-smallest reflections of R_0.
	
	By <Ref>, for each vertex b of C_0 there is a unique vertical elliptic isometry u ∈ [1,w] such that l(u) = n and u fixes b.
	By taking the left complement of these vertical elliptic isometries, we obtain that there are exactly n+1 vertical reflections r such that the right complement of r fixes a vertex of C_0.
	We only need to show that they are the n+1≺-smallest reflections of R_0.
	
	If w is a bipartite Coxeter element, then the n+1≺-smallest reflections are those that fix p_1 or p_2.
	By <Ref>, these are precisely the ones that have a right complement that fixes a vertex of C_0.
	
	Suppose now that w is a (p,q)-bigon Coxeter element in a Coxeter group W of type Ã_n.
	By part (i) of <Ref>, the n+1≺-smallest reflections of R_0 are those that fix one of p_1, p_2, …, p_m with m = p+q/(p,q).
	Let r ∈ R_0 be a vertical reflection such that its right complement fixes a vertex b of C_0.
	Let C be the axial chamber immediately below (r) ∩ℓ.
	By <Ref>, r is positive (otherwise its right complement would not fix a vertex of C_0) and $̱ is a vertex ofC.
	Suppose by contradiction thatris not among then+1≺-smallest reflections ofR_0.
	Then(r)intersectsℓin a pointp_iwithi ≥ m+1, and there are at leastm+1axial chambers betweenC_0andC(includingC_0andCthemselves).
	By part (ii) of <Ref>, the axial point$̱ is a vertex of exactly m consecutive axial chambers.
	This is a contradiction because $̱ is a vertex of bothC_0andC.




It is convenient to introduce some additional notation.
Given a simplexσ = [x_1|x_2|…|x_d] ∈ K_W, letπ(σ) = x_1x_2… x_d ∈ [1,w].
Also, ifis thed-fiber component containingσ, letλ(σ)(resp.ρ(σ)) be the simplex that appears immediately to the left (resp. immediately to the right) ofσin.
More explicitly:
rClλ(σ)    =   
		[x_1|…|x_d-1]    if π(σ) = w

		[y|x_1|… | x_d]    otherwise (here y is the left complement of x_1… x_d)
ρ(σ)    =   
		[x_2|…|x_d]    if π(σ) = w

		[x_1|… | x_d|y]    otherwise (here y is the right complement of x_1… x_d).
Thenλis the inverse ofρ.
More generally, we say that a simplexτ∈ K_Wis to the left of σ (resp. to the right of σ) ifτ = λ^k(σ)(resp.τ= ρ^k(σ)) for somek > 0.
Notice that, ifσandτbelong to a finite component, thenτis both to the left ofσand to the right ofσ.


With this notation, the definitions ofX_W'andK_W'can be written as follows:

  * σ∈(X_W') if and only if π(σ) fixes a vertex of C_0;
	
  * σ∈(K_W') if and only if λ^k(σ) ∈(X_W') for some k ≥ 0 and ρ^k(σ) ∈(X_W') for some k ≥ 0.




The following definition will be used in the construction of the matching on(K_W').


	Let σ = [x_1|x_2|…|x_d] ∈(K_W), with π(σ) = w.
	Define the depthδ(σ) of σ as the minimum i ∈{1,2,…, d} such that one of the following occurs:
	
  * l(x_i) ≥ 2;
		
  * l(x_i) = 1, i ≤ d-1, and x_i ≺ r for every reflection r≤ x_i+1 in [1,w].
	
	If no such i exists, let δ(σ) = ∞.
	
	Let σ∈(K_W').
	If π(σ) = w, and π(ρ(σ)) fixes a vertex of C_0, then δ(σ) ≠∞.
	
	Let σ = [x_1|x_2|…|x_d], and suppose by contradiction that δ(σ) = ∞.
	Then each x_i is a reflection, d = l(w) = n+1, and x_1 ≻ x_2 ≻…≻ x_n+1.
	Since π(ρ(σ)) = x_2… x_n+1 fixes a vertex of C_0, by <Ref> we have that x_1 is among the n+1≺-smallest reflections of R_0.
	Then x_1, x_2, …, x_n+1 are the n+1≺-smallest reflections of R_0.
	
	Let [y_1|y_2|…|y_n+1] be a (n+1)-simplex to the left of σ.
	Then y_1 = φ^j(x_i) for some i ∈{1, …, n+1 } and j < 0.
	The right complement y_2… y_n+1 of y_1 is equal to φ^j(u), where u is the right complement of x_i.
	Since x_i is among the n+1≺-smallest reflections, its right complement u fixes a vertex b of C_0 by <Ref>.
	Then y_2… y_n+1 = φ^j(u) fixes the axial vertex w^-j(b).
	By <Ref> (for the case Ã_n) and <Ref> (for the other cases), w^-j(b) is not a vertex of C_0.
	Therefore [y_2|…|y_n+1] ∉(X_W').
	This conclusion applies to every n-simplex [y_2|…|y_n+1] to the left of σ.
	Then σ∉(K_W'), which is a contradiction.



We are now ready to define a functionμwhich will ultimately give us the matching we need.



	Given a simplex σ∈(K_W') ∖(X_W'), define a simplex μ(σ) ∈(K_W) as follows.
	
  *  If π(σ) ≠ w, let μ(σ) = λ(σ).
		
		
  *  If π(σ) = w, and π(ρ(σ)) does not fix a vertex of C_0, let μ(σ) = ρ(σ).
	
	Suppose now that π(σ) = w, and π(ρ(σ)) fixes a vertex of C_0. Let σ = [x_1|x_2|…|x_d] and δ = δ(σ). Notice that δ≠∞ by <Ref>.
	
  *  If l(x_δ) ≥ 2, define μ(σ) = [x_1|…|x_δ-1|y|z|x_δ+1|…|x_d], where y is the ≺-smallest reflection of R_0 ∩ [1,x_δ], and yz = x_δ.
		
		
  *  If l(x_δ) = 1, define μ(σ) = [x_1|…|x_δ-1|x_δ x_δ+1|x_δ+2|… | x_d].
	
	Notice that μ(σ) ⋗σ if σ occurs in case (1) or (3), whereas μ(σ) ⋖σ if σ occurs in case (2) or (4).
	In addition, μ(σ) ∉(X_W') by <Ref>.
	[Matching for Ã_2]
	<Ref> shows the matching defined by μ on (K_W') ∖(X_W') in the case Ã_2, using the axial ordering of <Ref>.
	See also <Ref>, where the involved simplices are indicated by white nodes.
	The first 4 pairs on the left column occur in cases (1) and (2) of <Ref>.
	The other 6 pairs occur in cases (3) and (4).

	Let σ∈(K_W') ∖(X_W') be a simplex such that π(σ) = w, and ρ(σ) ∈(X_W').
	Then also ρ(μ(σ)) ∈(X_W').
	
	Let σ = [x_1|x_2|…|x_d].
	It is not possible for σ to occur in case (1) or (2) of <Ref>, because π(σ) = w and ρ(σ) ∈(X_W').
	
	If σ occurs in case (4), then we have μ(σ) = [x_1|…|x_δ-1|x_δ x_δ+1|x_δ+2|… | x_d].
	Therefore π(ρ(μ(σ))) ≤π(ρ(σ)) in [1,w].
	Since π(ρ(σ)) fixes a vertex of C_0, also π(ρ(μ(σ))) fixes a vertex of C_0 by <Ref>.
	Then ρ(μ(σ)) ∈(X_W').
	
	If σ occurs in case (3), then μ(σ) = [x_1|…|x_δ-1|y|z|x_δ+1|…|x_d].
	If δ > 1, we have π(ρ(μ(σ))) = π(ρ(σ)), and so π(ρ(μ(σ))) fixes a vertex of C_0.
	Assume from now on that δ = 1, so μ(σ) = [y|z|x_2|…|x_d].
	Let $̱ be a vertex ofC_0fixed byπ(ρ(σ)) = x_2… x_d.
	Letu ∈ [1,w]be the unique vertical elliptic element that fixes$̱ with l(u)=n (see <Ref>).
	By <Ref>, we have x_2… x_d ≤ u in [1,w].
	Let r be the left complement of u (it is a vertical reflection).
	By <Ref>, r is one of the n+1≺-smallest reflections of R_0.
	Passing to the left complements in the inequality x_2… x_d ≤ u, we get x_1 ≥ r in [1,w].
	By definition of μ, we have that y is the ≺-smallest reflection of R_0 ∩ [1,x_1], and therefore y ≼ r.
	In particular, y is one of the n+1≺-smallest reflections of R_0.
	By <Ref>, its right complement zx_2… x_d = π(ρ(μ(σ))) fixes a vertex of C_0.

	For every σ∈(K_W') ∖(X_W'), we have that μ(σ) ∈(K_W') ∖(X_W').
	
	We have already noted in <Ref> that μ(σ) ∉(X_W').
	We have μ(σ) = λ(σ) in case (1), and μ(σ) = ρ(σ) in case (2).
	Since σ∈(K_W') ∖(X_W'), in these two cases μ(σ) ∈(K_W').
	Suppose from now on that σ occurs in case (3) or case (4).
	In particular, π(σ) = w and ρ(σ) ∈(X_W').
	
	Let σ = [x_1|x_2|…|x_d].
	By <Ref>, we have ρ(μ(σ)) ∈(X_W').
	Therefore we only need to prove that there is a simplex in (X_W') to the left of μ(σ).
	Since σ∈(K_W'), there is a (d-1)-simplex τ∈(X_W') to the left of σ.
	It has the following form:
	
    τ = [φ^h(x_i+1) | … | φ^h(x_d) | φ^h+1(x_1) | … | φ^h+1(x_i-1)],

	for some h < 0 and i ∈{1,…, d}.
	
	If σ occurs in case (4), then μ(σ) = [x_1|…|x_δ-1|x_δ x_δ+1|x_δ+2|… | x_d] where δ = δ(σ).
	To the left of μ(σ) there is a simplex τ' such that π(τ') ≤π(τ), given by:
	
    [φ^h(x_δ+2) | … | φ^h(x_d) | φ^h+1(x_1) | … | φ^h+1(x_δ-1)]    if i=δ or i = δ+1.
    
    			[φ^h(x_i+1)|… | φ^h(x_δ x_δ+1) | … | φ^h(x_d)| φ^h+1(x_1) | … | φ^h+1(x_i-1)]    if i < δ.
    
    			[φ^h(x_i+1)|… | φ^h(x_d)| φ^h+1(x_1) | … | φ^h+1(x_δ x_δ+1) | … | φ^h+1(x_i-1)]    if i > δ + 1.

	Since τ∈(X_W'), we have that π(τ) fixes a vertex of C_0, so π(τ') also fixes a vertex of C_0 (by <Ref>), which means that τ' ∈(X_W').
	Therefore μ(σ) ∈(K_W').
	
	If σ occurs in case (3), then μ(σ) = [x_1|…|x_δ-1|y|z|x_δ+1|…|x_d].
	If i ≠δ, we can find a simplex τ' to the left of μ(σ) such that π(τ') = π(τ), namely:
	
    [φ^h(x_i+1) | … | φ^h(y) | φ^h(z) | … | φ^h(x_d) | φ^h+1(x_1) | … | φ^h+1(x_i-1)]    if i < δ
    
    			
    			[φ^h(x_i+1) | … | φ^h(x_d) | φ^h+1(x_1) | … | φ^h+1(y) | φ^h+1(z) | … | φ^h+1(x_i-1)]    if i > δ.

	As before, this implies that μ(σ) ∈(K_W').
	Suppose from now on that i = δ.
	
	The right complement of φ^h(x_δ) is equal to π(τ), which is elliptic because it fixes a vertex of C_0. Therefore φ^h(x_δ) is vertical, and thus also x_δ = yz is vertical.
	Let z = r_1r_2… r_m be the increasing factorization of z, with m ≥ 1.
	By definition of y, we have that y ≺ r_1 ≺…≺ r_m.
	Since yz is vertical, at least one of y, r_1, …, r_m is a vertical reflection.
	
  *  Case 1: r_m is a negative vertical reflection.
		By <Ref>, there exists a j < 0 such that φ^j(r_m) is among the n+1≺-smallest reflections of R_0.
		By <Ref>, its right complement u fixes a vertex of C_0.
		Consider the following simplex to the left of μ(σ):
		
    τ' = [φ^j(x_δ+1) | … | φ^j(x_d) | φ^j+1(x_1) | … | φ^j+1(x_δ-1) | φ^j+1(y)].

		The left complement of π(τ') is φ^j(z), and we have φ^j(r_m) ≤φ^j(z) in [1,w].
		Passing to the right complements in this inequality, we obtain that u ≥π(τ').
		Since u fixes a vertex of C_0, by <Ref> also π(τ') fixes a vertex of C_0.
		Therefore τ' ∈(X_W').
		
		
  *  Case 2: r_m is a horizontal reflection or a positive vertical reflection.
		The same is true for all the reflections y, r_1, …, r_m, because y ≺ r_1 ≺…≺ r_m.
		Recall that at least one of them is vertical.
		Then, for some k ∈{0, …, m}, we have that y, r_1, …, r_k are positive vertical reflections, and r_k+1, …, r_m are horizontal reflections.
		Since h < 0, we have that φ^h(y), φ^h(r_1), …, φ^h(r_k) are also positive vertical reflections.
		The compatibility property of ≺ then implies φ^h(y) ≺φ^h(r_1) ≺…≺φ^h(r_k).
		Let r_k+1'… r_m' be the increasing factorization of φ^h(r_k+1… r_m).
		Since r_k+1… r_m is horizontal, φ^h(r_k+1… r_m) is also horizontal, so the reflections r_k+1', …, r_m' are horizontal.
		Therefore we have φ^h(y) ≺φ^h(r_1) ≺…≺φ^h(r_k) ≺ r_k+1' ≺…≺ r_m'.
		By construction, the product of these m+1 reflections yields the increasing factorization of φ^h(yz) = φ^h(x_δ).
		In particular, φ^h(y) is the ≺-smallest reflection of [1, φ^h(x_δ)] by <Ref>.
		
		 Recall that τ∈(X_W'), so π(τ) fixes a vertex $̱ ofC_0.
		Letu ∈ [1,w]be a vertical elliptic element that fixes$̱ and such that l(u) = n (see <Ref>).
		By <Ref>, we have π(τ) ≤ u in [1,w].
		If we pass to the left complements in this inequality, we get φ^h(x_δ) ≥ r where r is the left complement of u (it is a vertical reflection).
		By <Ref>, r is among the n+1≺-smallest reflections of R_0.
		Since φ^h(y) is the ≺-smallest reflection of [1,φ^h(x_δ)], we have φ^h(y) ≼ r, and thus φ^h(y) is also among the n+1≺-smallest reflections of R_0.
		By <Ref>, the right complement of φ^h(y) fixes a vertex of C_0.
		Consider the following simplex to the left of μ(σ):
		
    τ' = [φ^h(z)|φ^h(x_δ+1)| … |φ^h(x_d)|φ^h+1(x_1)|… |φ^h+1(x_δ-1)].

		We have that π(τ') is the right complement of φ^h(y), so it fixes a vertex of C_0, and therefore τ' ∈(X_W'). 
	The function μ(K_W') ∖(X_W') →(K_W')∖(X_W') is an involution, i.e. it satisfies μ(μ(σ)) = σ.
	In addition, if σ occurs in case (3) or (4) of <Ref>, then δ(μ(σ)) = δ(σ).
	<Ref> shows that the image of μ is contained in (K_W') ∖(X_W'), so we can compose μ with itself.
	Let σ = [x_1|x_2|…|x_d] ∈(K_W') ∖(X_W').
	
	If σ occurs in case (1) of <Ref>, then μ(σ) = λ(σ) occurs in case (2), so μ(μ(σ)) = ρ(λ(σ)) = σ.
	Similarly, if σ occurs in case (2), then μ(σ) = ρ(σ) occurs in case (1), and μ(μ(σ)) = λ(ρ(σ)) = σ.
	
	If σ occurs in case (3) or (4), by definition of δ = δ(σ) we have that x_1, …, x_δ-1 are reflections such that x_1 ≻ x_2 ≻…≻ x_δ-1.
	In addition, x_δ-1≻ y if y is the ≺-smallest reflection of [1,x_δ].
	If σ occurs in case (3), then x_1 ≻…≻ x_δ-1≻ y, and y ≺ r for every reflection r ≤ z.
	Therefore δ(μ(σ)) = δ, and μ(σ) occurs in case (4), so μ(μ(σ)) = σ.
	If σ occurs in case (4), then δ(μ(σ)) = δ because x_δ-1≻ x_δ. In particular, μ(σ) occurs in case (3).
	By definition of δ, we also have that x_δ≺ r' for every reflection r' ≤ x_δ+1.
	Then, if we concatenate x_δ with the increasing factorization of x_δ+1, we get the increasing factorization of x_δ x_δ+1. Therefore x_δ is the ≺-smallest reflection of [1,x_δ x_δ+1], and μ(μ(σ)) = σ. 



Thanks to <Ref>, we can finally define a matchingon(K_W'):

    = { (μ(σ), σ) |σ∈(K_W') ∖(X_W')  and μ(σ) ⋖σ}.

A simplexσ∈(K_W')is critical if and only ifσ∈(X_W').
It only remains to prove thatis acyclic and proper.
For this, our strategy is to define a poset(P, ⊴)and
a mapξ(K_W') ∖(X_W') → Pthat decreases along alternating paths.

LetP ⊆ R_0^n+1be the set of all minimal length factorizations ofwas a product of reflections.
We endowPwith the total ordering⊴defined as follows.
Letα, α' ∈ P, and denote byr(resp.r') the≺-largest reflection appearing inα(resp.α').

  *  If r ≠ r', then set α⊲α' if and only if r ≻ r'.
	
	
  *  If r = r', let k (resp. k') be the position where r (= r') appears in α (resp. α').
	If k ≠ k', then set α⊲α' if and only if k > k'.
	
	
  *  If r = r' and k = k', then set α⊲α' if and only if α is lexicographically smaller than α' (as usual, reflections are compared using the total ordering ≺ of R_0).

The transitive property of⊴is immediate to check.

Define a closure operatorκ(K_W') ∖(X_W') →(K_W') ∖(X_W')in the following way:

    κ(σ) =
    σ   if π(σ) = w
    λ(σ) = μ(σ)    otherwise.

Given a simplexσ∈(K_W') ∖(X_W'), defineξ(σ) ∈ Pas the concatenation of the increasing factorizations ofx_1, x_2, …, x_d, where[x_1|x_2|…|x_d] = κ(σ).
Sinceπ(κ(σ)) = w, we have thatξ(σ)is indeed a minimal length factorization ofw.



	For every σ∈(K_W') ∖(X_W'), we have ξ(μ(σ)) = ξ(σ).
	
	Suppose without loss of generality that μ(σ) ⋖σ, so that σ occurs in case (2) or (4) of <Ref>.
	If σ occurs in case (2), then κ(μ(σ)) = κ(σ), and therefore ξ(μ(σ)) = ξ(σ).

	Suppose now that σ occurs in case (4).
	Then κ(σ) = σ and κ(μ(σ)) = μ(σ).
	Let σ = [x_1|x_2|…|x_d] and μ(σ) = [x_1|…|x_δ-1|x_δ x_δ+1|x_δ+2|…|x_d], where δ = δ(σ).
	As already noticed in the proof of <Ref>, the increasing factorization of x_δ x_δ+1 is given by the reflection x_δ followed by the increasing factorization of x_δ+1.
	Therefore ξ(μ(σ)) = ξ(σ).

	Let σ = [x_1|x_2|…|x_d] ∈(K_W') ∖(X_W') be a simplex such that π(σ) = w.
	There exist a negative vertical reflection r ∈ R_0 and an index i ∈{1,2,…, d} such that r ≤ x_i in [1,w].
	In particular, the ≺-largest reflection appearing in ξ(σ) is a negative vertical reflection.
	
	Since σ∈(K_W'), there is a (d-1)-simplex τ∈(X_W') to the left of σ.
	It has the following form:
	
    τ = [φ^h(x_i+1)| … | φ^h(x_d) | φ^h+1(x_1) | … | φ^h+1(x_i-1)],

	for some h < 0 and i ∈{1,…, d}.
	We have that π(τ) fixes a vertex $̱ ofC_0, andφ^h(x_i)is the left complement ofπ(τ).
	By <Ref>, there is a reflectionr' ≤φ^h(x_i)among then+1≺-smallest reflections ofR_0(see the last part of the proof of <Ref>).
	By <Ref>,r = φ^-h(r')is a negative vertical reflection, becauseh < 0.
	This proves the first part of the statement becauser ≤ x_i.
	
	The≺-increasing factorization ofx_iends with the≺-largest reflectionr”of[1,x_i]by <Ref>.
	Ifr̅is the≺-largest reflection appearing inξ(σ), we haver̅≽ r”≽ r, sor̅is a negative vertical reflection.

	Let σ, τ∈(K_W') ∖(X_W') be two simplices such that π(σ) = w and τ is a face of σ.
	Then ξ(τ) ⊴ξ(σ).
	If, in addition, τ = λ(σ), then we have the strict inequality ξ(τ) ⊲ξ(σ).
	
	Let σ = [x_1|x_2|…|x_d].
	Notice that κ(σ) = σ, because π(σ) = w.
	Let r (resp. r') be the ≺-largest reflection of ξ(τ) (resp. ξ(σ)), appearing in position k (resp. k').
	
	
  *  Case 1: τ = [x_2|…|x_d].
		Then τ = μ(σ) and therefore ξ(τ) = ξ(σ) by <Ref>.
		
		
  *  Case 2: τ= [x_1|… | x_i-1|x_ix_i+1|x_i+2|… |x_d] for some i ∈{1, …, d-1}.
		In particular, we have π(τ) = w and therefore κ(τ) = τ.
		Since [1, x_i] and [1, x_i+1] are both included in [1, x_ix_i+1], the ≺-largest reflection of [1, x_ix_i+1] is at least as ≺-large as the ≺-largest reflections of [1,x_i] and [1,x_i+1].
		Therefore r ≽ r'.
		If r ≻ r', then ξ(τ) ⊲ξ(σ), as desired.
		
		 Suppose that r = r'.
		The ≺-largest reflection of [1,x_ix_i+1] appears as the last reflection of the increasing factorization of x_ix_i+1.
		Therefore k ≥ k'.
		If k > k', then ξ(τ) ⊲ξ(σ), as desired.
		
		 By <Ref>, the increasing factorization of x_i x_i+1 is lexicographically smaller than (or equal to) the concatenation of the increasing factorizations of x_i and x_i+1.
		Then ξ(τ) is lexicographically smaller than (or equal to) ξ(σ).
		Therefore ξ(τ) ⊴ξ(σ).
		
		
  *  Case 3: τ = [x_1|…|x_d-1] = λ(σ).
		Then κ(τ) = [φ^-1(x_d)|x_1|…|x_d-1] ∈(K_W') ∖(X_W').
		
		 Suppose by contradiction that r ≺ r'.
		Since x_1, …, x_d-1 are common to both τ and σ, we have that r' ≤ x_d.
		Then φ^-1(r') ≤φ^-1(x_d), and therefore φ^-1(r') ≼ r ≺ r'.
		By <Ref>, r' is a negative vertical reflection.
		Then (φ^-1(r')) ∩ℓ is above (r') ∩ℓ.
		Since φ^-1(r') ≺ r', we have that φ^-1(r') is a positive vertical reflection.
		By <Ref>, φ^-1(r') is among the n+1≺-smallest reflections of R_0.
		By <Ref>, its right complement u fixes a vertex of C_0.
		Since φ^-1(r') ≤φ^-1(x_d), passing to the right complements we get u ≥ x_1… x_d-1 = π(τ).
		By <Ref>, (u) ⊆(π(τ)), and so π(τ) also fixes a vertex of C_0.
		This is a contradiction, because τ∉(X_W').
		Therefore r ≽ r'.
		If r ≻ r', then ξ(τ) ⊲ξ(σ), as desired.
		
		 
		Suppose now that r = r'.
		If r = r' ≤ x_d, the previous argument yields again a contradiction.
		Therefore r ≤ x_i for some i ∈{1, …, d-1}.
		Then the position k (where r appears in ξ(τ)), is strictly greater than the position k' (where r appears in ξ(σ)).
		Thus ξ(τ) ⊲ξ(σ). 
	The matching  on (K_W') is acyclic.
	
	Suppose by contradiction that there is an alternating cycle σ_1 ⋗τ_1 ⋖σ_2 ⋗τ_2 ⋖…⋗τ_m ⋖σ_m+1 = σ_1 in (K_W'), with m≥ 1.
	We have that (τ_j, σ_j) ∉ and (τ_j, σ_j+1) ∈ for all j ∈{1, …, m}.
	In particular, all the simplices involved are matched, so they are in (K_W') ∖(X_W').
	Also, by <Ref> we have that π(σ_j) = w for all j.
	
	By <Ref> and <Ref>, we have 
    ξ(σ_1) ⊵ξ(τ_1) = ξ(σ_2) ⊵ξ(τ_2) = …⊵ξ(τ_m) = ξ(σ_m+1) = ξ(σ_1).

	Then, all these inequalities are actually equalities:
	
    ξ(σ_1) = ξ(τ_1) = ξ(σ_2) = ξ(τ_2) = … = ξ(τ_m) = ξ(σ_m+1) = ξ(σ_1).

	By the second part of <Ref>, we have τ_j ≠λ(σ_j) for all j.
	Also, τ_j ≠ρ(σ_j) for all j, because otherwise we would have τ_j= μ(σ_j) i.e. (τ_j, σ_j) ∈.
	Since τ_j is a face of σ_j different from λ(σ_j) and ρ(σ_j), we have π(τ_j) = π(σ_j) = w for all j.
	As a consequence, each τ_j occurs in case (3) of <Ref>, and each σ_j occurs in case (4).
	
	By the second part of <Ref>, δ(τ_j) = δ(σ_j+1) for all j.
	In addition, since ξ(σ_j) = ξ(τ_j), we have δ(σ_j) ≥δ(τ_j) for all j.
	As before, since σ_m+1= σ_1, all inequalities are actually equalities:
	
    δ(σ_1) = δ(τ_1) = δ(σ_2) = δ(τ_2) = … = δ(τ_m) = δ(σ_m+1) = δ(σ_1).

	
	Let σ_1 = [x_1|x_2|…|x_d], τ_1 = [x_1|…|x_i-1|x_ix_i+1|x_i+2|…|x_d] for some i ∈{1, …, d-1}, and δ = δ(σ_1).
	Since σ_1 occurs in case (4), we have l(x_δ) = 1.
	Also, l(x_ix_i+1) ≥ 2 implies δ(τ_1) ≤ i.
	Since δ(τ_1) = δ(σ_1) = δ, we deduce that i ≥δ.
	If i = δ, then τ_1 = μ(σ_1), which is impossible because (τ_1, σ_1) ∉.
	Therefore i > δ, so τ_1 also occurs in case (4), because l(x_δ) = 1.
	However τ_1 occurs in case (3), and this is a contradiction.

	Let W be an irreducible affine Coxeter group, with a set of simple reflections S = {s_1, s_2, …, s_n+1} and a Coxeter element w = s_1s_2… s_n+1.
	The interval complex K_W deformation retracts onto its subcomplex X_W'.
	
	By <Ref> and <Ref>, K_W deformation retracts onto its ca­no­ni­cal nice subcomplex K_W'.
	We have constructed a matching  on the face poset (K_W').
	This matching has (X_W') as the set of critical cells and is acyclic by <Ref>.
	It is also proper because (K_W') is finite.
	By the main theorem of discrete Morse theory (<Ref>), K_W' deformation retracts onto X_W'.




We can finally prove theK(π,1)conjecture for affine Artin groups.


	Let W be an irreducible affine Coxeter group.
	The K(π,1) conjecture holds for the corresponding Artin group G_W.
	
	Fix a set of simple reflections S = {s_1, s_2, …, s_n+1} and a Coxeter element w = s_1s_2… s_n+1.
	By <Ref>, the interval complex K_W is a classifying space.
	By <Ref>, we have homotopy equivalences Y_W ≃ X_W' ≃ K_W, where Y_W is the orbit configuration space associated with W.
	Therefore Y_W is a classifying space for its fundamental group G_W.



We also obtain a new proof of the following theorem of McCammond and Sulway.



	Let W be an irreducible affine Coxeter group, with a set of simple reflections S = {s_1, s_2, …, s_n+1} and a Coxeter element w = s_1s_2… s_n+1.
	The natural homomorphism from the Artin group G_W to the dual Artin group W_w is an isomorphism.

	Consider the homotopy equivalences X_W ≃ X_W' ≃ K_W of <Ref>.
	The composition ψ X_W → K_W sends the 1-cell c_{s}, associated with a simple reflection s ∈ S, to the corresponding 1-cell [s] of K_W, preserving the orientation.
	Then the induced map ψ_* on the fundamental groups is exactly the natural homomorphism G_W → W_w, which is, therefore, an isomorphism.


§ THE FOUR INFINITE FAMILIES


The main purpose of this appendix is to prove <Ref>, <Ref>, <Ref> (these are statements aboutÃ_n), and <Ref> (for the casesÃ_n,B̃_n,C̃_n, andD̃_n).
We do this by explicitly examining the four infinite families of irreducible affine Coxeter groups.
This appendix can be also used as a source of examples, and it complements the computations of <cit.>.
We refer to <cit.> for the standard construction of root systems (see also <cit.>).





 §.§ Case An tilde


LetWbe a Coxeter group of typeÃ_n. It is realized as the reflection group inE = ^n+1 / ⟨ 1,…, 1 ⟩associated with the hyperplane arrangement

    = {{ x_i - x_j = k }| 1 ≤ i < j ≤ n+1,   k ∈}.

If∈^n+1, denote by[]its class inE = ^n+1/⟨ 1, …, 1⟩.

Let(p,q)be a pair of positive integers such thatp+q = n+1.
Label the coordinates of^n+1 = ^p ×^qas follows:x_1,x_2,…, x_p, y_1, y_2, …, y_q.
Given a point∈̱E, denote its coordinates byx_1^,̱…, x_p^,̱ y_1^,̱…, y_q^$̱ (they are well defined up to a multiple of (1, …, 1)).
Construct a (p,q)-bigon Coxeter element w as in <cit.>:

    w ()̱ = [x_p^+̱1, x_1^,̱…, x_p-1^|̱y_q^-̱1, y_1^,̱…, y_q-1^]̱.

Then the shortest vector in (w) is μ = [1/p, …, 1/p| -1/q, …, -1/q], and the points of the Coxeter axis ℓ (i.e. the points  such that w() =  + μ) are of the form

    [ p-1/p, p-2/p, …, 1/p, 0 | 0, 1/q, …, q-2/q, q-1/q] + θμ

for θ∈.
The hyperplanes of the form { x_i - x_j = k} or { y_i - y_j = k} are horizontal, whereas those of the form { x_i - y_j = k} are vertical.


	From (<ref>) it is immediate to see that the Coxeter axis ℓ is not contained in any hyperplane of .
	The value of θ that yields the intersection point of a vertical hyperplane { x_i -  y_j = k } with ℓ satisfies
	
    (1/p + 1/q)θ = k - p-i/p + j-1/q = k-1 + i/p + j-1/q.

	If we let k, i, and j vary, then θ can assume any value which is an integer multiple of (p,q)/p+q.


Consider now a point ∈ E which is not contained in any hyperplane of , and let C_ be the chamber containing .
In particular, for every j we have that y_j^ - x_p^∉, because otherwise  would lie on some vertical hyperplane { x_p - y_j = k }.
Consider the line ℓ_ passing through  and with the same direction as the Coxeter axis:

    ℓ_ = { [x_1^, x_2^, …, x_p^| y_1^, y_2^, …, y_q^] + θμ|θ∈}.

Let S_⊆ R be the set of the reflections with respect to the walls of C_.
Write S_ = S_^+ ⊔ S_^- ⊔ S_^, where S_^+ (resp. S_^-) consists of the reflections that intersect ℓ_ above (resp. below) , and S_^ consists of the horizontal reflections.



	Let W be a Coxeter group of type Ã_n, and w a (p,q)-bigon Coxeter element as in (<ref>).
	Let ∈ E be a point which is not contained in any hyperplane of , and such that x_p^ < x_p-1^ < … < x_1^ < x_p^ +1 and y_1^ < y_2^ < … < y_q^ < y_1^ + 1.
	Then the reflections in S_^+ (resp. S_^-) pairwise commute.
	In addition, w can be written as a product of the reflections in S_, where the reflections in S_^+ come first, and the reflections in S_^- come last.

Since the coordinates of  are defined up to a multiple of (1, …, 1), we can assume that x_p^= 0.
Therefore we have 0 = x_p^ < x_p-1^ < … < x_1^ < 1.
Let s be the (unique) index such that the fractional part of y_s^ is minimal, and let h = ⌊ y_s^⌋ be the largest integer which is less than y_s^.
Then we have

    h < y_s^ < y_s+1^ < … < y_q^ < y_1^ + 1 < … < y_s-1^ + 1 < h+1.


Let X = { x_1^, …, x_p^} and Y = { y_s^-h, …, y_q^ - h, y_1^ + 1 - h, …, y_s-1^ + 1 - h }.
The set Z = X ∪ Y consists of n+1 distinct real numbers between 0 (included) and 1 (excluded).
Write Z = { 0 = z_1^ < z_2^ < … < z_n+1^}, where each z_l stands either for some x_i or for some translate of some y_j (notice that z_1 = x_p).
Then the inequalities z_1 < z_2 < … < z_n+1 < z_1+1 define the chamber C_, and the walls of C_ are:

    { z_1 = z_2}, { z_2 = z_3 }, …, {z_n+1 = z_1 + 1 }.


For every i ∈{1,…, p}, the coordinate x_i appears in exactly two walls.
If they are both vertical, say { x_i - y_j = k} and {x_i - y_j+1 = k' }, then the first of these walls intersects ℓ_ below , because x^_i - y_j^ > k, whereas the second one intersects ℓ_ above , because x^_i - y_j+1^ < k' (here the indices of y are taken modulo q).
A similar argument applies to the coordinates y_j for j ∈{1, …, q}.
This proves that the reflections in S_^+ (resp. S_^-) pairwise commute.

The horizontal walls are among the following: { x_i = x_i+1} for 1 ≤ i ≤ p-1; {x_1 = x_p + 1}; {y_j = y_j+1} for 1 ≤ j ≤ q-1; {y_1 = y_q-1}.
Let t be the smallest index such that x_t^ < y_s^ - h.
Order the possible horizontal walls as follows:

    { x_t+1 = x_t+2},  { x_t+2 = x_t+3},  …,  {x_p-1 = x_p }, 
       { x_1 = x_p + 1 },  { x_1 = x_2 },  …,  { x_t-1 = x_t }, 
       { y_s = y_s+1},  { y_s+1 = y_s+2},  …,  { y_q-1 = y_q }, 
       { y_1 = y_q - 1 },  { y_1 = y_2 },  …,  { y_s-2 = y_s-1}.

Let w_+ (resp. w_-) be the product of the reflections in S_^+ (resp. S_^-), and let w_ be the product of the reflections in S_^, in the same relative order as in (<ref>).
Let ŵ = w_+ w_ w_-.
We want to prove that ŵ = w.
For this, it is enough to show that the linear part of ŵ coincides with the linear part of w, and that ŵ ()̱ = w()̱ for at least one point ∈̱E.

Denote by e_x_i (resp. e_y_j) the unit vector in the direction of x_i (resp. y_j).
Given two elements ζ, ζ' ∈ Z, we write ζ⋖ζ' if ζ < ζ' and there exists no ζ”∈ Z between ζ and ζ'.
For j ∈{1,…, q} define

    k_j = 
    	-h    if j ≥ s
    
    	1-h    othewise,

so that Y consists of the real numbers y_j^ + k_j.

Consider the unit vector e_x_i with i ≤ p-1.


  *  If x_i+1^⋖ x_i^, then: the linear part of w_- fixes e_x_i; the linear part of w_ sends e_x_i to e_x_i+1; the linear part of w_+ fixes e_x_i+1.

	
  *  If there is at least one element of Z between x_i+1^ and x_i^, say x_i+1^⋖ y_j^ + k_j ⋖ y_j+1^ + k_j+1⋖…⋖ y_j'^ + k_j'⋖ x_i^, then: the linear part of w_- sends e_x_i to e_y_j'; the linear part of w_ sends e_y_j' to e_y_j; the linear part of w_+ sends e_y_j to e_x_i+1.

Consider now the unit vector e_x_p.

  *  If x_1^ is the maximal element of Z, then: the linear part of w_- fixes e_x_p; the linear part of w_ sends e_x_p to e_x_1; the linear part of w_+ fixes e_x_1.
	
	
  *  Otherwise, if x_1^⋖ y_j^ + k_j ⋖ y_j+1^ + k_j+1⋖…⋖ y_s-1^ + k_s-1, then: the linear part of w_- sends e_x_p to e_y_s-1; the linear part of w_ sends e_y_s-1 to e_y_j; the linear part of w_+ sends e_y_j to e_x_1.

A similar argument shows that the linear part of ŵ sends e_y_j to e_y_j+1 if j ≤ q-1, and sends e_y_q to e_y_1.
Therefore the linear part of ŵ coincides with the linear part of w (see (<ref>)).

It remains to show that ŵ()̱ = w()̱ for some point $̱.
Let$̱ be the vertex of C_ opposite to the wall H = { x_t - y_s = -h }.
The point $̱ is the intersection of all the other walls ofC_, so its coordinates are explicitly determined by the following equations:
rCl3lx_p^=̱ x_p-1^=̱… = x_t^

	y_s^-̱ h    =    y_s+1^-̱ h = … = y_q^-̱ h = y_1^+̱ 1 -h = …= y_s-1^+̱ 1 -h 
   =    x_t-1^=̱ x_t-2^=̱… = x_1^=̱ x_p^+̱ 1.

Therefore,

    =̱ [ 1, …, 1_t-1, 0, …, 0_p-t+1|h, …, h_s-1, h+1, …, h+1_q-s+1 ].

Recall that the reflectionrwith respect toHbelongs toS_^+.
Since$̱ is fixed by all the other reflections in S_, we have that

    ŵ()̱ = r()̱ = [ 1, …, 1_t, 0, …, 0_p-t|h, …, h_s, h+1, …, h+1_q-s ] = w()̱.

	Let  be a point in C ∩ℓ, as in (<ref>).
	<Ref> immediately implies points (ii) and (iii).
	
	Since p ≥ q, none of the walls of C is of the form {y_j - y_j' = k}.
	In addition, for every j ∈{1,…, q}, the coordinate y_j appears in the defining equation of exactly two walls of C.
	The corresponding reflections are one in S^+ and one in S^-.
	Therefore |S^+| = |S^-| = q and |S^| = n+1 - 2q = p-q, proving point (i).

	Let d = (p,q).
	For part (i), write  as in (<ref>) for some θ∈.
	Let {x_i - y_j = k} be a vertical hyperplane containing .
	In particular, we have x_i^ - y_j^∈.
	For another hyperplane { x_i' - y_j' = k' } to contain , we need x_i'^ - y_j'^∈.
	By (<ref>), we have that
	
    (x_i^ - y_j^) - (x_i'^ - y_j'^) = ( p-i+θ/p - j-1-θ/q) - ( p-i'+θ/p - j'-1-θ/q) = i'-i/p + j'-j/q.

	This is an integer if and only if i' = i + u ·p/d and j' = j + v ·q/d for some u, v ∈ such that d | u+v.
	There are d such pairs (i',j') ∈{1,…,p}×{1, …, q}, and each of them yields exactly one hyperplane containing .
	All these hyperplanes pairwise commute, because no two such pairs share the same i' or the same j'.
	
	For part (ii), fix a point ∈ C ∩ℓ.
	Let H = { x_i -y_j = k } be a vertical wall of C, and denote by $̱ the intersection point ofHwithℓ.
	By the description of the walls ofCgiven in the proof of <Ref>, we have eitherx_i^ < y_j^ + k < x_i^ + 1/porx_i^ - 1/p < y_j^ + k < x_i^.
	In any case,|x_i^ - y_j^ - k| < 1/p.
	Sinceb ∈ H, we also havex_i^-̱ y_j^-̱ k = 0, and therefore|(x_i^ - y_j^) - (x_i^-̱ y_j^)̱| < 1/p.
	Then, if we writeand$̱ as in (<ref>) for some θ_a, θ_b ∈, we obtain that |θ_a - θ_b| < q/p+q.
	By <Ref>, consecutive points of the sequence {p_i}_i ∈ differ by (p,q)/p+qμ.
	Therefore there are q/(p,q) = m possible positions for $̱ above, andmpossible positions below.
	More precisely, ifis between_iand_i+1, then$̱ must be one of the following 2m points: _i-m+1, _i-m+2, …, _i+m.
	By part (i), each of these points is contained in exactly (p,q) vertical hyperplanes of .
	By <Ref>, the chamber C has exactly 2q = 2m ·(p,q) vertical walls.
	Therefore every hyperplane of  that intersects ℓ in one of the previous 2m points must be a wall of C.

	We begin with part (i).
	Every point ∈ℓ satisfies x_i^ - 1 ≤ x_i'^≤ x_i^ for every i < i', and y_j^≤ y_j'^≤ y_j^+1 for every j < j'.
	Then the same non-strict inequalities need to be satisfied by every axial vertex $̱.
	The walls of an axial chamberChave the form (<ref>), so every vertex ofCadmits an expression where all the coordinates are integers.
	Therefore every axial vertex has the following form:
	
    b = [1, …, 1_p_1, 0, …, 0_p_2|h, …, h_q_1, h+1, …, h+1_q_2 ]

	for someh ∈, withq_2 ≥ 1(otherwise we can changehwithh+1).
	LetA ⊆ Ebe the set of points of the form (<ref>).
	The representation (<ref>) of a pointb ∈ Abecomes unique, and we call it the standard form ofb, if we also imposep_2 ≥ 1(otherwise we can remove1from all coordinates).
	
	By (<ref>), the Coxeter elementwacts onb ∈ Aas follows:
	
    w(b) = [1, …, 1_p_1+1, 0, …, 0_p_2-1|h, …, h_q_1+1, h+1, …, h+1_q_2-1 ].

	In particularw(b) ∈ A, so there is an action ofonAby powers ofw.
	Ifp_2 = 1, then we can remove1from all coordinates in the previous equation to obtain the standard form
	
    w(b) = [0, …, 0 |h-1, …, h-1_q_1+1, h, …, h_q_2-1 ].

	
	Denote byσ(b)the sum of the coordinates of the standard form of a pointb ∈ A.
	We have thatσ(w(b)) ≡σ(b)modulop+q.
	By looking at the walls ofCgiven by (<ref>), we see that thep+qvertices of an axial chamberChave distinct values ofσ, modulop+q.
	Therefore there are at leastp+qdifferent orbits for the action ofonA, and the vertices of an axial chamber belong to different orbits.

	It remains to show that there are exactlyp+qorbits.
	For every pointb ∈ A, there is a pointw^j(b)which is in one of the following forms:
	
  * [1, …, 1, 0 |0, …, 0_q_1, 1, …, 1_q_2 ] with q_2 ≥ 1;
		
		
  * [1, …, 1_p_1, 0, …, 0_p_2| 0, …, 0 ] with p_2 ≥ 1.
	
	There areqpoints of the form (a), andppoints of the form (b).
	Therefore there are exactlyp+qorbits.
	Notice that this also proves that every point ofAis an axial vertex because it is in the orbit of some axial vertex.
	
	To prove part (ii), fix an axial vertex∈̱A.
	Without loss of generality, we may assume that$̱ is of the form (a) or (b).
	We want to describe the set of points ∈ℓ that are contained in some axial chamber C having $̱ as one of its vertices.
	Letbe as in (<ref>), for someθ∈.
	
	If$̱ is of the form (a), then x_p^<̱ y_j^$̱ forj ≥ q_1+1, andy_j^<̱ x_i^$̱ for i ≤ p-1 and j ≤ q_1.
	The same strict inequalities need to be satisfied by , an so we get θ/p < j-1-θ/q for j ≥ q_1 + 1, and j-1-θ/q < p-i+θ/p for i ≤ p-1 and j ≤ q_1.
	These conditions are equivalent to q_1-1/q - 1/p < (1/p + 1/q) θ < q_1/q, so θ belongs to an interval of length 1.
	Conversely, every value of θ in this interval yields a point  contained in a chamber C which has $̱ as one of its vertices, provided that we exclude the finite set of values corresponding to points that belong to some vertical hyperplane of.
	By <Ref>, this interval of values ofθspans exactlyp+q/(p,q)axial chambers.
	
	If$̱ is of the form (b), the procedure is similar.
	We have y_j^<̱ x_i^$̱ for alljand fori ≤ p_1, andy_j^>̱ x_i^-̱ 1for alljand fori ≥ p_1+1.
	The corresponding inequalities for the pointgive the conditionp_1/p -1/q≤( 1/p + 1/q) θ≤p_1/p + 1/p, which again yields an interval of length1.



In order to prove <Ref>, we start by explicitly describing the hyperbolic elementsu ∈ [1,w]withl(u) = n.
These are obtained asu = wrwherer ∈ [1,w]is a horizontal reflection.
By <Ref>, the hyperplanes corresponding to horizontal reflections in[1,w]have the following forms:

  * { x_i = x_i'} for i < i';
	
  * { x_i = x_i' + 1} for i < i';
	
  * { y_j = y_j'} for j < j';
	
  * { y_j = y_j' - 1} for j < j'.


Ifris the reflection with respect to{ x_i = x_i'}withi < i' ≤ p-1, thenu = wrsends a point∈̱Eto

    [x_p^+̱1, x_1^,̱…, x_i-1^,̱ x_i'^,̱ x_i+1^,̱…, x_i'-1^,̱ x_i^,̱ x_i'+1^,̱…, x_p-1^|̱y_q^-̱1, y_1^,̱…, y_q-1^]̱.

On the coordinatesx_i+1, …, x_i', the hyperbolic isometryuacts as a horizontal Coxeter element of typeA_i'-i-1.
On the remaining coordinates, it acts as a(p-i'+i,q)-bigon Coxeter element of typeÃ_n-i'+i.
This is exactly the hyperbolic-horizontal decomposition ofu(<Ref>).

Ifris the reflection with respect to{x_i = x_p }for somei ≤ p-1, thenu=wrsends∈̱Eto

    [x_i^+̱1, x_1^,̱…, x_i-1^,̱ x_p^,̱ x_i+1^,̱…, x_p-1^|̱y_q^-̱1, y_1^,̱…, y_q-1^]̱.

As before, on the coordinatesx_i+1, …, x_pwe have thatuacts as a horizontal Coxeter element of typeA_p-i-1, and on the remaining coordinates it acts as a(i, q)-bigon Coxeter element of typeÃ_n-p+i.

Ifris the reflection with respect to{x_i = x_i'+1}withi < i' ≤ p-1, thenu=wrsends∈̱Eto

    [x_p^+̱1, x_1^,̱…, x_i-1^,̱ x_i'^+̱ 1, x_i+1^,̱…, x_i'-1^,̱ x_i^-̱ 1, x_i'+1^,̱…, x_p-1^|̱y_q^-̱1, …, y_q-1^]̱.

Thenuacts as a(i'-i,q)-bigon Coxeter element of typeÃ_q+i'-i-1on the coordinatesx_i+1, …, x_i', y_1, …, y_q, and as a horizontal Coxeter element of typeA_p-i'+i-1on the remaining coordinates.

Ifris the reflection with respect to{x_i = x_p + 1}for somei ≤ p-1, thenu=wrsends∈̱Eto

    [x_i^,̱ x_1^,̱…, x_i-1^,̱ x_p^+̱ 1, x_i+1^,̱…, x_p-1^|̱y_q^-̱1, y_1^,̱…, y_q-1^]̱.

As before, on the coordinatesx_i+1, …, x_p, y_1, …, y_qwe have thatuacts as a(p-i, q)-bigon Coxeter element of typeÃ_n-i, and on the remaining coordinates it acts as a horizontal Coxeter element of typeA_i-1.

A similar phenomenon happens ifris a reflection of type (iii) or (iv).
We are now ready to prove <Ref>.


	Let u ∈ [1,w] be a hyperbolic isometry such that W_u is irreducible.
	By iterating the previous argument, we get that u acts as a (p',q')-bigon Coxeter element of type Ã_p'+q'-1 on a subset x_i_1, …, x_i_p', y_j_1, …, y_j_q' of the coordinates (for some i_1 < … < i_p' and j_1 < … < j_q').
	It acts as the identity on the other coordinates, because otherwise u would have a non-trivial hyperbolic-horizontal decomposition, and W_u would be reducible.
	If we restrict to the relevant p'+q' coordinates, we get
	
    u()̱ = [x_i_p'^+̱ 1, x_i_1^,̱…, x_i_p'-1^|̱y_j_q'^-̱ 1, y_j_1^,̱…, y_j_q'-1^]̱.

	Then W_u is a Coxeter group of type Ã_p'+q'-1.
	
	The point  of the statement can be written in the form (<ref>), and its relevant p'+q' coordinates are given by
	
    [ p-i_1/p, p-i_2/p, …, p-i_p'/p| j_1-1/q, j_2-1/q, …, j_q'-1/q] + θ [1/p,…, 1/p| -1/q, …, -1/q].

	In particular, notice that x_i_p'^ < x_i_p'-1^ < … < x_i_1^ < x_i_p'^ + 1 and y_j_1^ < y_j_2^ < … < y_j_q'^ < y_j_1^ + 1.
	We conclude by applying <Ref> to the Coxeter group W_u (in place of W), its Coxeter element u (in place of w), and the point .


 §.§ Case Cn tilde


LetWbe a Coxeter group of typeC̃_n.
It is realized as the reflection group inE = ^nassociated with the hyperplane arrangement

    = {{x_i ± x_j = k }|  1 ≤ i < j ≤ n,   k ∈}∪{{x_i = k/2}|  1 ≤ i ≤ n,   k ∈}.


Consider the chamberC_0 = { 0 < x_1 < x_2 < … < x_n < 1/2}, with walls given by{x_1 = 0}, {x_1 = x_2}, …, {x_n-1 = x_n}, { x_n = 1/2}.
Letwbe the Coxeter element obtained by multiplying the reflections with respect to the walls ofC_0(in the order given before).
Thenwacts as follows on a point∈̱E:

    w()̱ = (x_n^-̱ 1, x_1^,̱…, x_n-1^)̱.

The shortest vector in(w)isμ = -( 1/n, 1/n, …, 1/n), and the points of the Coxeter axisℓare of the form

    (  0, 1/n, 2/n, …, n-1/n) + θμ

forθ∈.
The hyperplanes of the form{ x_i - x_j = k}are horizontal, and the other hyperplanes ofare vertical.


Given a point∈ Ewhich is not contained in any hyperplane of, letC_be the chamber containing, and consider the line passing throughand with the same direction as the Coxeter axis:

    ℓ_ = { (x_1^, x_2^, …, x_n^) + θμ|θ∈}.

DefineS_, S_^+, S_^-, S_^as in the caseÃ_n(Section <ref>).


	Let W be a Coxeter group of type C̃_n, and w a Coxeter element as in (<ref>).
	Let ∈ E be a point which is not contained in any hyperplane of , and such that x_1^ < x_2^ < … < x_n^ < x_1^ +1.
	Then the reflections in S_^+ (resp. S_^-) pairwise commute.
	In addition, w can be written as a product of the reflections in S_, where the reflections in S_^+ come first, and the reflections in S_^- come last.

	The statement holds for  if and only if it holds for w^m(), for any m ∈.
	Notice that w permutes cyclically the fractional parts of the real numbers x_1^, …, x_n^.
	Therefore, without loss of generality, we can assume that x_1^ has the smallest fractional part among x_1^, …, x_n^.
	Since w^n is a pure translation of nμ = -(1, 1,…, 1), we can also assume that 0 < x_1^ < 1.
	Together with the hypothesis x_1^ < x_2^ < … < x_n^ < x_1^ +1, we obtain that
	
    0 < x_1^ < x_2^ < … < x_n^ < 1.

	
	Let p ∈{0,…, n} be the largest index such that x_p^ < 1/2.
	Let q = n-p, and define y_j = 1 - x_p+j for j ∈{1,…, q}.
	Notice that the isometry
	
    (x_1, …, x_n) ↦ (x_1, …, x_p, 1-x_p+1, …, 1-x_n)

	is an element of W, so it sends chambers to chambers.
	Since  is not contained in any hyperplane of , the new coordinates x_1^, …, x_p^, y_1^, …, y_q^ are pairwise distinct numbers between 0 and 1/2.
	They satisfy x_1^ < … < x_p^ and y_1^ > … > y_q^.
	Let Z = {x_1^, …, x_p^, y_1^, …, y_q^}, and write Z = { z_1^ < z_2^ < … < z_n^}, where each z_l is either equal to some x_i or to some y_j.
	The inequalities 0 < z_1 < z_2 < … < z_n < 1/2 define the chamber C_, and the walls of C_ are
	
    { z_1 = 0}, { z_1 = z_2}, …, { z_n-1 = z_n}, { z_n = 1/2}.

	The vertical walls that intersect ℓ_ above  are those of the form { y_j = x_i } with y_j^⋖ x_i^, { x_1 = 0 }, and { y_1 = 1/2}.
	Notice that not all of these hyperplanes necessarily occur as walls of C_.
	For example, {x_1 = 0} is a wall of C_ if and only if x_1^ < y_q^.
	Similarly, the vertical walls that intersect ℓ_ below  are those of the form { x_i = y_j } with x_i^⋖ y_j^, { y_q = 0 }, and { x_p = 1/2}.
	Every coordinate x_i or y_j appears in exactly two walls. If these two walls are both vertical, then one intersects ℓ_ above  and the other intersects ℓ_ below .
	Therefore, the reflections in S_^+ (resp. S_^-) pairwise commute.
	
	The horizontal walls are among the following: { x_i = x_i+1} for 1 ≤ i ≤ p-1; {y_j = y_j+1} for 1 ≤ j ≤ q-1.
	Notice that we left out the hyperplanes {x_1 = x_p - 1} and { y_1 = y_q + 1 }: although the corresponding reflections are in [1,w], they cannot occur as walls of C_, by (<ref>).
	Order these hyperplanes as follows:
	
    { x_1 = x_2 },  { x_2 = x_3 },  …,  {x_p-1 = x_p }, 
       { y_1 = y_2 },  { y_2 = y_3 },  …,  {y_q-1 = y_q }.

	Let w_+ (resp. w_-) be the product of the reflections in S_^+ (resp. S_^-), and let w_ be the product of the reflections in S_, in the same relative order as in (<ref>).
	Let ŵ = w_+ w_ w_-.
	We want to prove that ŵ = w.
	
	If p=0 or q=0, the set of walls (<ref>) can be written explicitly, and it is immediate to check that ŵ = w (in the case q=0, the chamber C_ is precisely the one used to define w in the first place).
	Suppose from now on that p > 0 and q > 0.
	We are going to show that the linear parts of ŵ and w coincide, and that ŵ()̱ = w()̱ for some point ∈̱E.
	
	Consider the unit vector e_x_i in the direction of x_i, with i ≤ p-1.
	
  *  If x_i^⋖ y_j^⋖ y_j-1^⋖…⋖ y_j'^⋖ x_i+1^ with j' ≤ j, then: the linear part of w_- sends e_x_i to e_y_j; the linear part of w_ sends e_y_j to e_y_j'; the linear part of w_+ sends e_y_j' to e_x_i+1.
		
  *  If x_i^⋖ x_i+1^, then: the linear part of w_- fixes e_x_i; the linear part of w_ sends e_x_i to e_x_i+1; the linear part of w_+ fixes e_x_i+1.
	
	Consider now the unit vector e_x_p.
	
  *  If x_p^⋖ y_j^⋖ y_j-1^⋖…⋖ y_1^ < 1/2 with j ≥ 1, then: the linear part of w_- sends e_x_p to e_y_j; the linear part of w_ sends e_y_j to e_y_1; the linear part of w_+ sends e_y_1 to -e_y_1.
		
		
  *  If y_1^⋖ x_i^⋖ x_i+1^⋖…⋖ x_p^ < 1/2 with i ≤ p, then: the linear part of w_- sends e_x_p to -e_x_p; the linear part of w_ sends -e_x_p to -e_x_i; the linear part of w_+ sends -e_x_i to -e_y_1.
	
	A similar argument shows that the linear part of ŵ sends e_y_j to e_y_j+1 if j ≤ q-1, and sends e_y_q to -e_x_1.
	Recall now that y_j = 1-x_p+j, and so e_y_j = -e_x_p+j.
	Therefore the linear part of ŵ sends e_x_i to e_x_i+1 for all i ∈{1,…, n} (with indices taken modulo n), so it coincides with the linear part of w.
	
	It remains to show that ŵ()̱ = w()̱ for some point ∈̱E.
	
  *  If 0 < x_1^ < y_q^, let $̱ be the vertex ofC_opposite to the wallH = { x_1 = 0}, i.e.=̱(1/2, 1/2, …, 1/2).
		The reflectionrwith respect toHbelongs toS_^+.
		Thenŵ()̱ = r()̱ = (-1/2, 1/2, …, 1/2) = w()̱.
		
		
  *  If0 < y_q^⋖ y_q-1^⋖…⋖ y_j^⋖ x_1^withj ≤ q, let$̱ be the vertex of C_ opposite to the wall H = { y_j = x_1 } = { 1-x_p+j = x_1 }, i.e.
		
    =̱( 1/2, …, 1/2_p+j-1, 1, …, 1_q-j+1).

		Again, the reflection r with respect to H belongs to S_^+, and therefore
		
    ŵ()̱ = r()̱ = ( 0, 1/2, …, 1/2_p+j-1, 1, …, 1_q-j) = w()̱.



As in the caseÃ_n, in order to prove <Ref> we start by explicitly describing the hyperbolic elementsu ∈ [1,w]withl(u) = n.
By <Ref>, the horizontal reflectionsr ∈ [1,w]are the ones with the following fixed hyperplanes:

  * { x_i = x_j} for i < j;
	
  * { x_i = x_j - 1} for i < j.


Ifris the reflection with respect to{ x_i = x_j }withi < j ≤ n-1, thenu = wrsends a point∈̱Eto

    (x_n^-̱ 1, x_1^,̱…, x_i-1^,̱ x_j^,̱ x_i+1^,̱…, x_j-1^,̱ x_i^,̱ x_j+1^,̱…, x_n-1^)̱.

On the coordinatesx_i+1, …, x_j, the hyperbolic isometryuacts as a horizontal Coxeter element of typeA_j-i-1. On the remaining coordinates, it acts as a Coxeter element of typeC̃_n-j+i.

Ifris the reflection with respect to{ x_i = x_n }, thenu=wrsends∈̱Eto

    (x_i^-̱ 1, x_1^,̱…, x_i-1^,̱ x_n^,̱ x_i+1^,̱…, x_n-1^)̱.

Thereforeuacts as a horizontal Coxeter element of typeA_n-i-1on the coordinatesx_i+1, …, x_n, and as a Coxeter element of typeC̃_ion the coordinatesx_1, …, x_i.

The situation is similar ifris a reflection with respect to{ x_i = x_j - 1}for somei < j.
In this case,uacts as a Coxeter element of typeC̃_j-ion the coordinatesx_i+1, …, x_j, and as a horizontal Coxeter element of typeA_n-j+i-1on the remaining coordinates.

Notice that, in some of the previous cases, a Coxeter element of typeC̃_1can occur (for instance, this happens ifris the reflection with respect to{x_1 = x_n}).
The limit caseC̃_1still makes sense and coincides withÃ_1.



	Let u ∈ [1,w] be a hyperbolic isometry such that W_u is irreducible.
	By the same argument as in the case Ã_n, we have that u acts as a Coxeter element of type C̃_m on a subset x_i_1, …, x_i_m of the coordinates, and as the identity on the remaining coordinates.
	If we restrict to the relevant coordinates x_i_1, …, x_i_m, we have
	
    u()̱ = (x_i_m^-̱ 1, x_i_1^,̱…, x_i_m-1^)̱.

	The relevant coordinates of  are given by
	
    = ( i_1-1/n, i_2-1/n, …, i_m-1/n) - θ (1/n,1/n,…, 1/n)

	for some θ∈.
	These coordinates satisfy x_i_1^ < x_i_2^ < … < x_i_m^ < x_i_1^ + 1.
	We conclude by applying <Ref> to the Coxeter group W_u, its Coxeter element u, and the point .


 §.§ Case Bn tilde


LetWbe a Coxeter group of typeB̃_n.
It is realized as the reflection group inE = ^nassociated with the hyperplane arrangement

    = {{x_i ± x_j = k }|  1 ≤ i < j ≤ n,   k ∈}∪{{x_i =  k }|  1 ≤ i ≤ n,   k ∈}.


Consider the chamberC_0 = { 0 < x_1 < x_2 < … < x_n,   x_n-1 + x_n < 1 }, with walls given by{x_1 = 0}, {x_1 = x_2}, …, {x_n-1 = x_n}, { x_n-1 + x_n = 1 }.
Letwbe the Coxeter element obtained by multiplying the reflections with respect to these walls.
Thenwacts as follows on a point∈̱E:

    w()̱ = (x_n-1^-̱ 1, x_1^,̱…, x_n-2^|̱1 - x_n^)̱.

The shortest vector in(w)isμ = -( 1/n-1, 1/n-1, …, 1/n-1| 0 ), and the points of the Coxeter axisℓare of the form

    (  0, 1/n-1, 2/n-1, …, n-2/n-1| 1/2) + θμ

forθ∈.
The horizontal hyperplanes are:{ x_i - x_j = k}for1 ≤ i < j ≤ n-1andk ∈;{ x_n = k}fork ∈.

Given a point∈ Ewhich is not contained in any hyperplane of, defineC_,S_,S_^+,S_^-, andS_^as in the previous cases.



	Let W be a Coxeter group of type B̃_n, and w a Coxeter element as in (<ref>).
	Let ∈ E be a point which is not contained in any hyperplane of , and such that x_1^ < x_2^ < … < x_n-1^ < x_1^ +1 and x_n^= 1/2.
	Then the reflections in S_^+ (resp. S_^-) pairwise commute.
	In addition, w can be written as a product of the reflections in S_, where the reflections in S_^+ come first, and the reflections in S_^- come last.

	By replacing  with a w^m() for a suitable m ∈, we can assume that 0 < x_1^ < x_2^ < … < x_n-1^ < 1 and x_n^ = 1/2.
	Let p ∈{0, …, n-1} be the largest index such that x_p^ < 1/2, and let q = n-1-p.
	As in the case C̃_n, define y_j = 1 - x_p+j for j ∈{1, …, q}.
	Define also
	
    t = 
    	x_n    if q is even
    
    	1-x_n    if q is odd.

	Notice that, if we multiply the reflections with respect to { x_i = x_n } and { x_i + x_n = 1 } (for some i ≤ n-1), we obtain the isometry (x_i, x_n) ↦ (1-x_i, 1-x_n).
	If we multiply these isometries for all i ∈{p+1, …, n-1}, we obtain the change of coordinates
	
    (x_1, …, x_n-1| x_n) ↦ (x_1, …, x_p, y_1, …, y_q | t),

	which is therefore an element of W.
	As in the case C̃_n, we now have 0 < x_1^ < x_2^ < … < x_p^ < 1/2 and 0 < y_q^ < y_q-1^ < … < y_1^ < 1/2.
	In addition, there is the last coordinate t^ = 1/2.
	
	Let Z = { x_1^, …, x_p^, y_1^, …, y_q^}, and write Z = { z_1^ < z_2^ < … < z_n-1^}.
	Using the coordinates z_1, …, z_n-1, t, the chamber C_ is given by { 0 < z_1 < … < z_n-1 < t,   z_n-1 + t < 1 }.
	Therefore, its walls are
	
    { z_1 = 0 }, { z_1 = z_2 }, …, {z_n-1 = t}, { z_n-1 + t = 1}.

	
	Denote by r and r' the reflections with respect to { z_n-1 = t } and {z_n-1 + t = 1}, respectively.
	They commute, and they are both vertical.
	In addition, they are either both in S_^+ (if z_n-1 = y_1) or both in S_^- (if z_n-1 = x_p).
	The product rr' is given by (z_n-1, t) ↦ (1-z_n-1, 1-t), and it is the identity on the other coordinates.
	Since t is either x_n or 1-x_n, we have that rr' is given by (z_n-1, x_n) ↦ (1-z_n-1, 1-x_n).
	On the last coordinate x_n, this is exactly how w acts.
	On the coordinate z_n-1, we have that rr' acts as a reflection with respect to z_n-1 = 1/2.
	The rest of the proof carries out exactly as in the case C̃_n-1 (see <Ref>).



Let us examine the hyperbolic elementsu ∈ [1,w]withl(u) = n.
The horizontal reflectionsr ∈ [1,w]are:

  * { x_i = x_j} for i < j ≤ n-1;
	
  * { x_i = x_j - 1} for i < j ≤ n-1;
	
  * { x_n = 0 };
	
  * { x_n = 1 }.


Similarly to the caseC̃_n, ifris a reflection of type (i), thenu = wracts as a horizontal Coxeter element of typeA_j-i-1on some of the coordinates, and as a Coxeter element of typeB̃_n-j+ion the remaining coordinates.
Ifris a reflection of type (ii), thenu=wracts as a Coxeter element of typeB̃_j-i+1on some of the coordinates, and as a horizontal Coxeter element of typeA_n-j+i-2on the remaining coordinates.
Notice that the limit caseB̃_2 = C̃_2can arise (for instance, ifris the reflection with respect to{x_1 = x_n-1}).

Ifris the reflection with respect to{x_n = 0}, thenu = wrsends a point∈̱Eto

    u()̱ = (x_n-1^-̱ 1, x_1^,̱…, x_n-2^|̱x_n^+̱ 1).

This is a(n-1,1)-bigon Coxeter element of typeÃ_n-1, andW_uis a Coxeter group of typeÃ_n-1.
Ifris the reflection with respect to{x_n = 1}we obtain the same result up to a conjugation byw, sou = wris again a(n-1,1)-bigon Coxeter element of typeÃ_n-1.


	Let u ∈ [1,w] be a hyperbolic element such that W_u is irreducible.
	In particular, u = w v^-1 for some horizontal element v ∈ [1,w].
	
	If r ≤ v, where r is a reflection with respect to { x_n = 0} or {x_n = 1}, then u ≤ wr. Since wr is a Coxeter element of type Ã_n-1, this case was already covered in Section <ref>.
	
	Otherwise we proceed as for the case C̃_n, and notice that u must act as a Coxeter element of type B̃_m on a subset of the coordinates x_i_1, …, x_i_m-1, x_n.
	If we restrict to these relevant coordinates, we have
	
    u()̱ = (x_i_m-1^-̱ 1, x_i_1^,̱…, x_i_m-2^|̱1 - x_n^)̱.

	We conclude by applying <Ref> to the Coxeter group W_u, its Coxeter element u, and the point .


 §.§ Case Dn tilde


LetWbe a Coxeter group of typeD̃_n.
It is realized as the reflection group inE = ^nassociated with the hyperplane arrangement

    = {{x_i ± x_j = k }|  1 ≤ i < j ≤ n,   k ∈}.


Consider the chamberC_0 = { 0 < x_1 + x_2,   x_1 < x_2 < … < x_n,   x_n-1 + x_n < 1 }, with walls given by{x_1 + x_2 = 0}, {x_1 = x_2}, …, {x_n-1 = x_n}, { x_n-1 + x_n = 1 }.
Letwbe the Coxeter element obtained by multiplying the reflections with respect to these walls.
Thenwacts as follows on a point∈̱E:

    w()̱ = (-x_1^|̱x_n-1^-̱ 1, x_2^,̱ x_3^,̱…, x_n-2^|̱1 - x_n^)̱.

The shortest vector in(w)isμ = -( 0 | 1/n-2, 1/n-2, …, 1/n-2| 0 ), and the points of the Coxeter axisℓare of the form

    ( 0 |  0, 1/n-2, 2/n-2, …, n-3/n-2| 1/2) + θμ

forθ∈.
The horizontal hyperplanes are:{ x_i - x_j = k}for2 ≤ i < j ≤ n-1andk ∈;{ x_1 ± x_n = k}fork ∈.

Given a point∈ Ewhich is not contained in any hyperplane of, defineC_,S_,S_^+,S_^-, andS_^as in the previous cases.



	Let W be a Coxeter group of type D̃_n, and w a Coxeter element as in (<ref>).
	Let ∈ E be a point which is not contained in any hyperplane of , and such that x_1^= 0, x_2^ < x_3^ < … < x_n-1^ < x_2^ +1, and x_n^= 1/2.
	Then the reflections in S_^+ (resp. S_^-) pairwise commute.
	In addition, w can be written as a product of the reflections in S_, where the reflections in S_^+ come first, and the reflections in S_^- come last.

	By replacing  with a w^m() for a suitable m ∈, we can assume that x_1^= 0, 0 < x_2^ < x_3^ < … < x_n-1^ < 1, and x_n^ = 1/2.
	Let p ∈{1, …, n-1} be the largest index such that x_p^ < 1/2, and let q = n-1-p.
	Define y_j = 1 - x_p+j for j ∈{1, …, q}, and
	
    t = 
    	x_n    if q is even
    
    	1-x_n    if q is odd.

	As in the case B̃_n, the change of coordinates
	
    (x_1 | x_2, …, x_n-1| x_n) ↦ (x_1 | x_2, …, x_p, y_1, …, y_q | t)

	is an element of W.

	We now have 0 = x_1^ < x_2^ < … < x_p^ < t^ = 1/2 and 0 = x_1^ < y_q^ < y_q-1^ < … < y_1^ < t^= 1/2.
	Let Z = { x_2^, …, x_p^, y_1^, …, y_q^}, and write Z = { z_2^ < z_3^ < … < z_n-1^}.
	Using the coordinates x_1, z_2, …, z_n-1, t, the chamber C_ is given by { 0 < x_1 + z_2,   x_1 < z_2 < … < z_n-1 < t,   z_n-1 + t < 1 }.
	Therefore its walls are
	
    { x_1 + z_2 = 0 }, { x_1 = z_2 }, {z_2 = z_3}, …, {z_n-1 = t}, { z_n-1 + t = 1}.

	
	Exactly as in the case B̃_n, the reflections with respect to { z_n-1 = t } and {z_n-1 + t = 1} commute, and they are either both in S_^+ or both in S_^-.
	Their product acts as (z_n-1, x_n) ↦ (1-z_n-1, 1-x_n).
	Similarly, the reflections with respect to { x_1 + z_2 = 0 } and { x_1 = z_2 } commute, and they are either both in S_^+ or both in S_^-.
	Their product acts as (x_1, z_2) ↦ (-x_1, -z_2).
	On the coordinate x_1, this is exactly how w acts.
	On the coordinate z_2, this is the same as a reflection with respect to z_2 = 0.
	We conclude as in the case B̃_n.



We now examine the hyperbolic elementsu ∈ [1,w]withl(u) = n.
The horizontal reflectionsr ∈ [1,w]are:

  * { x_i = x_j} for 2 ≤ i < j ≤ n-1;
	
  * { x_i = x_j - 1} for 2 ≤ i < j ≤ n-1;
	
  * { x_1 ± x_n = 0 };
	
  * { x_n ± x_1 = 1 }.


Similarly to the previous cases, ifris a reflection of type (i), thenu = wracts as a horizontal Coxeter element of typeA_j-i-1on some of the coordinates, and as a Coxeter element of typeD̃_n-j+ion the remaining coordinates.
Ifris a reflection of type (ii), thenu=wracts as a Coxeter element of typeD̃_j-i+2on some of the coordinates, and as a horizontal Coxeter element of typeA_n-j+i-3on the remaining coordinates.
Notice that the limit caseD̃_3 = Ã_3can arise (for instance, ifris the reflection with respect to{x_2 = x_n-1}). When this happens, a(2,2)-bigon Coxeter element is obtained.

Ifris the reflection with respect to{x_1 + x_n = 0}, thenu = wrsends a point∈̱Eto

    u()̱ = (x_n^|̱x_n-1^-̱ 1, x_2^,̱ x_3^,̱…, x_n-2^|̱x_1^+̱ 1).

This is a(n-2,2)-bigon Coxeter element of typeÃ_n-1, andW_uis a Coxeter group of typeÃ_n-1.
Ifris the reflection with respect to{x_1 + x_n = 1}we obtain the same result up to a conjugation byw, sou = wris again a(n-2,2)-bigon Coxeter element of typeÃ_n-1.

Ifris the reflection with respect to{x_n - x_1 = 0}, thenu = wrsends a point∈̱Eto

    u()̱ = (-x_n^|̱x_n-1^-̱ 1, x_2^,̱ x_3^,̱…, x_n-2^|̱1- x_1^)̱.

If we setx_1' = -x_1, using the coordinates(x_1', x_2, …, x_n)we get


    u()̱ = (x_n^|̱x_n-1^-̱ 1, x_2^,̱ x_3^,̱…, x_n-2^|̱x_1'^  + 1),

which is now clearly recognizable as a(n-2,2)-bigon Coxeter element of typeÃ_n-1.
Ifris the reflection with respect to{x_n - x_1 = 1}we obtain the same result up to a conjugation byw, sou = wris again a(n-2,2)-bigon Coxeter element of typeÃ_n-1.


	Let u ∈ [1,w] be a hyperbolic element such that W_u is irreducible.
	In particular, u = w v^-1 for some horizontal element v ∈ [1,w].
	
	If r ≤ v, where r is a reflection with respect to { x_1 ± x_n = 0} or {x_n ± x_1 = 1}, then u ≤ wr. Since wr is a Coxeter element of type Ã_n-1, this case was already covered in Section <ref>.
	
	Otherwise we proceed as for the cases C̃_n and B̃_n, by applying <Ref> to the Coxeter group W_u, its Coxeter element u, and the point .
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