# Le Conte de la Mesure sur les Complexes Cubiques CAT(0)

Talia Fernós  Department of Mathematics, University of North Carolina,
Greensboro<EMAIL_ADDRESS>In honor of Mike Mihalik’s 70th
birthday, happy birthday!

(Date: March 10, 2024)

###### Abstract.

We revisit the topic of probability measures on CAT(0) cube complexes and
prove that an amenable group acting on a CAT(0) cube complex, regardless of
dimension, necessarily preserves an interval in the Roller compactification.
In the finite dimensional case, we prove that there must be an orbit of
cardinality $2^{N}$, where $N$ is bounded by the dimension. This is a slight
extension of the author’s previuos Tits’ Alternative.

###### Key words and phrases:

CAT(0), Cube Complexes, Tits’ Alternative

###### 2020 Mathematics Subject Classification:

Primary 54C40, 14E20; Secondary 46E25, 20C20

###### 2020 Mathematics Subject Classification:

Primary 20F65, 20E05, 60B05; Secondary 05C12, 05E18, 06A07, 20F29

The author was partially supported by NSF grant DMS–2005640.

Introduction The idea for this short paper came through preparations for a
talk I gave in summer of 2023 at the Séminaire Virtuel Francophone – Groupes
et Géométrie. The seminar is aimed to be expository in nature and so I thought
I would revisit the topic of probability measures on CAT(0) cube complexes,
and their associated intervals (see Section 0.6). This association was
introduced by Monod and Shalom in [MS04, Proposition 4.5] for trees and
developed further in my joint work with Chatterji and Iozzi [CFI16]. It was
also used either implicitly or explicitly in [Fer18, FLM18, FLM21], and even
[FFH23]. The key tool for this is the Lifting Decomposition which is discussed
in Section Lemma.

As I was preparing my lecture, I realized that I could in fact make a small
improvement on my previous contribution to the story of Tits’ Alternatives.
Therefore, I take the opportunity to share that here. I also include some of
the drawings I made (using notability on an ipad).

######  Theorem.

Let $X$ be a CAT(0) cube complex of finite dimension $D$, and suppose that
$\Gamma\to\text{Aut}\,X$ preserves an interval in the Roller compactification
$\overline{X}$. Then $\Gamma$ must have an orbit in $\overline{X}$ of
cardinality $2^{N}$, for some $0\leqslant N\leqslant D$.

If $X$ is not finite dimensional and $\Gamma$ is assumed to be amenable then
it must preserve an interval $\mathcal{I}\subset\overline{X}$.

Combining with the Tits’ Alternative from [Fer18], we obtain the following
(the reader should compare statements):

######  Corollary (Tits’ Alternative).

Suppose $X$ is a finite dimensional CAT(0) cube complex. Given a group action
$\Gamma\to\text{Aut}\,X$ either $\Gamma$ contains a freely acting free group
on 2 generators, or $\Gamma$ has an orbit of cardinality $2^{N}$ in
$\overline{X}$, for some $0\leqslant N\leqslant D$, where $D$ is the dimension
of $X$.

Examples: The following examples show that the bounds above are optimal. We
note that all the groups are amenable.

  * ❀

Consider the standard action of the infinite dihedral group
$\mathrm{D}_{\infty}$ on $\mathbb{Z}$. Of course, there is no fixed point but
there is an orbit of cardinality 2 in the Roller compactification, namely
$\\{\pm\infty\\}$.

  * ❀

Similarly, $\mathrm{D}_{\infty}^{N}$ acts on $\mathbb{Z}^{N}$ and has a unique
finite orbit which has cardinality $2^{N}$.

  * ❀

Finally, $\underset{n\in\mathbb{N}}{\overset{}{\oplus}}\mathrm{D}_{\infty}$
acts on $\underset{n\in\mathbb{N}}{\overset{}{\oplus}}\mathbb{Z}$ which is of
course infinite dimensional, and does not have a finite orbit in the Roller
compactification, but it is an interval.

We note that there are _many_ Tits’ Alternatives available in the literature
[Tit72, BFH04, SW05, CS11, Fio18, MP22, OP22, GJN23, Gen21]. However, in this
brief note, I will not endeavor to give a comprehensive overview of the Tits’
Alternatives, nor to give a thorough account of CAT(0) cube complexes. The
reader can see the references provided for further details.

Acknowledements: I thank Dafne Sanchez for help with the coloring scheme,
inspired by Matisse. I also thank Indira Chatterji, François Dahmani, Anne
Lonjou, and Yves Stalder for the invitation to speak at the Séminaire Virtuel
Francophone – Groupes et Géométrie which inspired me to write this article. I
thank all of my collaborators and mentors with whom I have discussed CAT(0)
cube complexes for helping me explore how to understand and communicate their
beauty. Finally, I thank the National Science Foundation for their generous
support of my work, particularly through NSF grant DMS–2005640. Basics

A cube complex is a space obtained by gluing unit cubes isometrically along
their faces. Moreover, it is said to be CAT(0) if it is nonpositively curved
and simply connected.

You may think of the process of creating a CAT(0) cube complex as gluing
together cubes in a particular order. If at some point there is positive
curvature that is created locally, it is due to something as in Figure 1.
There, we have created part of the boundary of a 3-dimensional cube by gluing
3 unit squares together. That creates a cone angle of $3\pi/2$. Since
$3\pi/2<2\pi$, it is positively curved. However, we may annul the positive
curvature by “filling” in the other side to create a 3-dimensional cube.
Nonpositive curvature of a CAT(0) cube complex is equivalent to the Gromov
Link Condition. We refer the reader to [Sag95, CN05, Nic04, Rol16] for more
details.

Figure 1. Making a cube complex CAT(0)

##  0.1. Functoriality: Walled Spaces

A useful way to think of CAT(0) cube complexes is via this functorial
construction. Begin with a set $Y$ together with a collection of (nonempty)
two-sided (walls) partitions $\mathcal{P}\subset 2^{Y}$. Two sided here means
that if $h\in\mathcal{P}$ then $Y\setminus h\in\mathcal{P}$. The wall may be
thought of as the pair $\\{h,Y\setminus h\\}$.

Next, create a graph by declaring that each maximal non-empty intersection of
sets from $\mathcal{P}$ is a vertex. We connect two vertices if their
corresponding defining intersections differ by the choice of exactly one side
of one partition from $\mathcal{P}$. In Figure 2, you can see a choice of 5
partitions. Each region is not empty. The regions labeled $A$ and $U_{1}$
differ by the choice of one partition and therefore, their associated vertices
$a$ and $u_{1}$ respectively are connected by an edge.

Figure 2. Functorial Construction

Once we have done this, we may not have a CAT(0) cube complex. However, we may
take the cubical completion (as in Figure 2, where we must add the unlabled
“back” vertex to creat a 3-cube) to annul positive curvature. We note that
this can always be done, except that distances may become infinite (if there
are infinitely many partitions separating two regions, i.e. the finite
interval condition is not satisfied) or the dimension may become infinite (if
there are families of pairwise transverse partitions of unbounded cardinality,
see Section 0.3 for the definition of transverse).

Conversely, start with (the vertex set of) a CAT(0) cube complex $X$. Each
edge in $X$ belongs to an equivalence class generated by “being parallel
across a square”. The compliment of that parallelism class has two sides, and
the vertices that belong to each side give the two-sided partition of the
vertex set (see Figure 3).

Finally, if we apply the previous construction to this collection of two-sided
partitions, we get a CAT(0) cube complex, which is cannonically isomorphic to
(the vertex set of) $X$.

Figure 3. Two CAT(0) Cube Complexes with 10 halfspaces

##  0.2. The Roller Compactification

A _halfspace_ of a CAT(0) cube complex is one side of a two-sided partition,
as discussed above. We have established that the vertex set, denoted by $X$,
is cannonically determined by the halfspace structure $\mathfrak{H}\subset
2^{X}$. By mapping a vertex in $X$ to the collection of halfspaces that
contain it, we obtain an isometric injection:

$X\hookrightarrow 2^{\mathfrak{H}}.$

Here the extended metric on $2^{\mathfrak{H}}$ is given by half the
cardinality of the symmetric difference $d(S,T)=\frac{1}{2}(S\triangle
T)\in[0,\infty]$. While this metric is useful, we will rely on the standard
topology on $2^{\mathfrak{H}}\cong\mathrm{Map}(\mathfrak{H}\to\\{0,1\\})$.
This is given by declaring the basic open sets to be cylinder sets, which are
themeselves determined by specifying values in _finitely many coordinates_.
Equivalently, this is the topology of pointwise convergence of maps
$\mathfrak{H}\to\\{0,1\\}$. With this topology, $2^{\mathfrak{H}}$ is compact.
We note that the induced topology may be different than the metric topology on
$X$. This is directly comparable to the weak-$*$ topology and respectively the
metric topology on a Hilbert space.

Having found an injection of $X$ into the compact space $2^{\mathfrak{H}}$, we
may take the closure in the image and this defines the _Roller
compactification_. By removing the image of $X$ inside this closure, we are
left with the _Roller boundary_.

Note that once we obtain the closure of $X$ in $2^{\mathfrak{H}}$, there is a
cannonical extension of the halfspaces as partitions of $X$ to $\overline{X}$.
Those are the basic clopen sets for our totally disconnected topology on
$\overline{X}$.

The following is then immediate from the functorial construction, and can be
thought of as forgetfulness.

######  Corollary.

Let $X$ be a CAT(0) cube complex with halfspace structure $\mathfrak{H}$. Let
$\mathfrak{H}^{\prime}\subset\mathfrak{H}$ be involution invariant and
$X^{\prime}$ the associated CAT(0) cube complex. Then the map
$2^{\mathfrak{H}}\twoheadrightarrow 2^{\mathfrak{H}^{\prime}}$, given by
$S\mapsto S\cap\mathfrak{H}^{\prime}$ induces a 1-Lipschitz projection
$\overline{X}\twoheadrightarrow\overline{X^{\prime}}$.

We shall see in Section Lemma how and when we can find a cannonical section to
this map. The example in Figure 4 is given by taking
$\mathfrak{H}^{\prime}=\mathfrak{H}\setminus\mathcal{B}$ and a valence
consideration shows that an isometric section is impossible.

Figure 4. An example where a section to the projection is impossible

The Roller boundary of a tree is, as a set, the visual boundary, but the
topology is different when it is not (large-scale) locally finite. Consider
the tree given by identifying at $0$ infinitely many copies of $[0,\infty]$,
as in Figure 5. It is not difficult to see that $\infty_{n}\to 0$ since all
halfspaces eventually contain $0$.

The Roller compactification has an important multiplicative property:

$\overline{X_{1}\times X_{2}}=\overline{X}_{1}\times\overline{X}_{2}.$

Figure 5. Halfspaces containing $\infty_{n}$.
Figure 6. The Roller Compactification for $\mathbb{Z}^{2}$
Figure 7. Roller Compactification for $F_{2}\times\mathbb{Z}$

Combining this, with the discussion for trees, we obtain the Roller
compactification of $\mathbb{Z}^{2}$, as in Figure 6, or for
$F_{2}\times\mathbb{Z}$ as in Figure 7.

##  0.3. Comparing halfspaces

Given a pair of halfspaces $h,k\in\mathfrak{H}$ such that $k\neq h,h^{*}$ then
either all of the pairwise intersections $h\cap k,h\cap k^{*},h^{*}\cap
k,h^{*}\cap k^{*}$ are not empty (and in this case, we say that $h$ and $k$
are _transverse_ , and write $h\pitchfork k$) or one of the following other
cases hold (see Figure 8):

$\displaystyle k\subset h\quad\,\,$ $\displaystyle\quad\,\,h\subset k$
$\displaystyle h^{*}\cap k^{*}=\varnothing$ $\displaystyle h\cap
k=\varnothing.$

Figure 8. The possible relationships between halfspaces $h,k$, with $k\neq
h,h^{*}$.

######  Lemma.

The dimension of $X$ is the supremum cardinality of subsets
$S\subset\mathfrak{H}$ of pairwise transverse halfspaces.

It is straightforward to verify that if
$\mathfrak{H}=\mathfrak{H}_{1}\sqcup\mathfrak{H}_{2}$ is a nontrivial disjoint
decomposition into nonempty involution invariant, pairwise transverse sets and
$X$, $X_{1}$, and $X_{2}$ are the assocaited CAT(0) cube complexes
respectively then there is a cannonical isomorphism $X\cong X_{1}\times
X_{2}$. We say that $X$ is irreducible if it is not isomorphic to a product.

######  Theorem.

[CS11, Proposition 2.6] Let $X$ be finite dimensional. Then, there is a
cannonical decomposition of $X$ into a product of irreducible CAT(0) cube
complexes.

A pair of halfspaces $h,k\in\mathfrak{H}$, with $h\neq k^{*}$ are said to be
_facing_ if $h^{*}\cap k^{*}=\varnothing$. This corresponds to the lower right
corner in Figure 8. A triple of halfspaces $h,k,\ell\in\mathfrak{H}$ are said
to be a _facing triple_ if they are pairwise facing.

##  0.4. Intervals in $\overline{X}$

Let $\mathcal{I}\subset\overline{X}$ be an arbitrary subset. The (involution
invariant) collection of halfspaces that separate points in $\mathcal{I}$ is

$\mathfrak{H}_{\mathcal{I}}:=\\{h\in\mathfrak{H}:h\cap\mathcal{I},h^{*}\cap\mathcal{I}\neq\varnothing\\}.$

Similarly, we denote the collection of halfspaces that contain $\mathcal{I}$
as $\mathfrak{H}_{\mathcal{I}}^{+}$, and the collection of compliments of
halfspaces that contain $\mathcal{I}$, namely the halfspaces that trivially
intersect $\mathcal{I}$ is denoted by $\mathfrak{H}_{\mathcal{I}}^{-}$,
meaning that
$\mathfrak{H}_{\mathcal{I}}^{-}=(\mathfrak{H}_{\mathcal{I}}^{+})^{*}$. We have
established the following decomposition:

$\mathfrak{H}=\mathfrak{H}_{\mathcal{I}}\sqcup\mathfrak{H}_{\mathcal{I}}^{+}\sqcup\mathfrak{H}_{\mathcal{I}}^{-}.$

We say that, $\mathcal{I}$ is an _interval_ if there exists
$x,y\in\overline{X}$ such that
$\mathcal{I}=\underset{h\in\mathfrak{H}_{\\{x,y\\}}^{+}}{\overset{}{\cap}}h$.
In this case we write $\mathcal{I}=\mathcal{I}(x,y)$ and say that it is the
_interval between $x$ and $y$_. We also call $x$ and $y$ _endpoints_ of
$\mathcal{I}$. Clearly, $\mathfrak{H}_{\mathcal{I}(x,y)}$ does not contain a
facing triple.

The structure of intervals in an arbitrary CAT(0) cube complex can be quite
exotic; please see the examples in Figures 9 and 10 or come up with your own!

Nevertheless, the following theorem of [BCG+09], which relies on Dilworth’s
Theorem for partially ordered sets, shows that intervals are not too wild.

######  Theorem.

Let $D$ be the dimension of $X$. If $\mathcal{I}\subset\overline{X}$ is an
interval then there exists an isometric embedding
$\mathcal{I}\hookrightarrow\overline{\mathbb{Z}^{D}}$.

######  Corollary.

If $\mathcal{I}\subset\overline{X}$ is an interval then the set of end points
of $\mathcal{I}$ has cardinality $2^{N}$, for some $0\leqslant N\leqslant D$.

Figure 9. The shaded region is an interval in the ambient CAT(0) cube complex
Figure 10. This Roller compactification is an interval

##  0.5. The Helly Property in $\overline{X}$

The following is an important property of halfspaces in a CAT(0) cube complex.
The first item is the standard Helly property for halfspaces. The second
follows from the first by applying the finite intersection property to the
collection of sets $\mathcal{S}$, each of which is clopen in the compact space
$\overline{X}$. Comparing it with the classical Helly property for convex sets
in euclidean space, we may consider it as telling us that, in a sense, CAT(0)
cube complexes are of Helly-dimension 1.

######  Lemma.

Suppose that $\mathcal{S}\subset\mathfrak{H}$ is such that $h\cap
k\neq\varnothing$ for all $h,k\in\mathcal{S}$.

  * •

If $\\#\mathcal{S}<\infty$ then
$\underset{h\in\mathcal{S}}{\overset{}{\cap}}h\cap X\neq\varnothing$.

  * •

Otherwise
$\overline{X}\supseteq\underset{h\in\mathcal{S}}{\overset{}{\cap}}h\neq\varnothing.$

Lifting Decompositions

Let $X$ be a CAT(0) cube complex with associated halfspace structure
$\mathfrak{H}$. We shall say $\mathcal{S}\subset\mathfrak{H}$ is _consistent_
if it satisfies the following two properties:

  1. (1)

If $h\in\mathcal{S}$ then $h^{*}\notin\mathcal{S}$.

  2. (2)

If $h,k\in\mathfrak{H}$, with $h\subset k$, and $h\in\mathcal{S}$ then
$k\in\mathcal{S}$.

We note that, according to the functorial construction, if $\mathcal{S}$ is
consistent and $\mathcal{S}\sqcup\mathcal{S}^{*}=\mathfrak{H}$ then
$\mathcal{S}$ is the collection of halfspaces containing a single point.

######  Proposition.

Let $X$ be a CAT(0) cube complex with associated halfspaces $\mathfrak{H}$ and
$\mathcal{S}\subset\mathfrak{H}$ consistent. Set
$\mathfrak{H}_{\mathcal{S}}:=\mathfrak{H}\setminus(\mathcal{S}\sqcup\mathcal{S}^{*})$
and let $X(\mathfrak{H}_{\mathcal{S}})$ be the associated CAT(0) cube complex.
Then the map $2^{\mathfrak{H}_{\mathcal{S}}}\to 2^{\mathfrak{H}}$ given by
$E\mapsto E\sqcup\mathcal{S}$ induces an isometric injection
$\overline{X}(\mathfrak{H}_{\mathcal{S}})\hookrightarrow\overline{X}$, whose
image is exactly

$\underset{h\in\mathcal{S}}{\overset{}{\cap}}\,h\subset\overline{X}.$

Furthermore, if $\mathcal{S}$ is $\Gamma$-invariant, for some action
$\Gamma\to\text{Aut}\,(X)$ then with the restricted action on the image, the
above natural injection is $\Gamma$-equivariant.

##  0.6. Probability Measures

Consider a probability measure $\mu\in\textrm{Prob}(\overline{X})$ and the
associated collection of halfspaces
$\mathfrak{H}_{\mu}^{+}:=\\{h\in\mathfrak{H}:\mu(h)>1/2\\}$. It is
straightforward to verify that $\mathfrak{H}_{\mu}^{+}$ is consistent and that
the collection of halfspaces
$\mathfrak{H}_{\mu}=\\{h\in\mathfrak{H}:\mu(h)=1/2\\}$ does not contain any
facing triples. Applying the Lifting Decomposition to
$\mathfrak{H}_{\mu}^{+}$, we get that
$\underset{h\in\mathfrak{H}^{+}_{\mu}}{\overset{}{\cap}}\,h$ is isomorphic to
$\overline{X}(\mathfrak{H}_{\mu})$.

######  Lemma.

[CFI16, Lemma 4.7] If $\mu\in\textrm{Prob}(\overline{X})$ then
$\overline{X}(\mathfrak{H}_{\mu})$ is an interval.

##  0.7. Medians

Let $x,y,z\in\overline{X}$ and consider the associated probability measure
$\mu_{(x,y,z)}=\frac{1}{3}(\delta_{x}+\delta_{y}+\delta_{z})$. A simple parity
argument shows that if $h\in\mathfrak{H}$ then
$\mu_{(x,y,z)}(h)\neq\frac{1}{2}$. Therefore, we have that
$\underset{h\in\mathfrak{H}_{\mu}^{+}}{\overset{}{\cap}}h$ is a single point,
which we shall call the _median_ of the triple and denote it by $m(x,y,z)$.

While this is not the standard definition of the median, it fits nicely within
our context. We note that several of the natural properties of the median
(e.g. invariance under permutation of the points) are immediate, including the
property that

$m(x,x,y)=x.$

Proof of the Main Theorem

Suppose $\Gamma$ is amenable acting on $X$. Since $\overline{X}$ is compact
and metrizable, there must be a $\Gamma$-invariant probability measure
$\mu\in\textrm{Prob}(\overline{X})$. By the Lifting Decomposition and the
previous lemma, it follows that
$\underset{h\in\mathfrak{H}_{\mu}^{+}}{\overset{\,}{\cap}}h$ is a
$\Gamma$-invariant interval.

Suppose now that $X$ is of finite dimension $D$ and that $\Gamma$ is not
necessarily amenable but preserves an interval
$\mathcal{I}\subset\overline{X}$. By Corollary 0.4 (see also [Fer18, Corollary
2.9]), the number of end points on which $\mathcal{I}$ is an interval is
$2^{D^{\prime}}$ for some $0\leqslant{D^{\prime}}\leqslant D$ and we identify
these with
$\left({\mathbb{Z}/2\mathbb{Z}}\right)^{D^{\prime}}\cong\\{0,1\\}^{D^{\prime}}$.
By [CS11, Proposition 2.6] we have that
$\text{Aut}\,(\\{0,1\\}^{D^{\prime}})\cong\\{0,1\\}^{D^{\prime}}\rtimes\mathrm{Sym}({D^{\prime}})$.
We may therefore project $\Gamma\to\text{Aut}\,(\\{0,1\\}^{D^{\prime}})$, and
without loss of generality, assume
$\Gamma\leqslant\text{Aut}\,(\\{0,1\\}^{D^{\prime}})$.

Let $\Gamma_{0}=\Gamma\cap\\{0,1\\}^{D^{\prime}}\lhd\Gamma$ and note that
$|\Gamma_{0}|=2^{N}$ for some $0\leqslant N\leqslant{D^{\prime}}$. Fix a
choice of right $\Gamma_{0}$-coset representatives $S\subset\Gamma$, with
trivial $\\{0,1\\}^{D^{\prime}}$-coordinate. Let
$\mathcal{O}\in\\{0,1\\}^{D^{\prime}}$ be defined component-wise by
$\mathcal{O}_{i}\equiv 0$, for $i\in\\{1,\dots,D^{\prime}\\}$. Note that
$\mathcal{O}$ is fixed by $\mathrm{Sym}({D^{\prime}})$ and has trivial
stabilizer in $\Gamma_{0}$. Let $(\mathcal{O},\alpha)\in S$. Then
$\Gamma_{0}(\mathcal{O},\alpha).\mathcal{O}=\Gamma_{0}.\mathcal{O}$. We have
shown that

$\Gamma.\mathcal{O}=\underset{(\mathcal{O},\alpha)\in
S}{\sqcup}\Gamma_{0}(\mathcal{O},\alpha).\mathcal{O}=\Gamma_{0}.\mathcal{O}$

Therefore, by the orbit stabilizer theorem, we have

$|\Gamma.\mathcal{O}|=|\Gamma_{0}.\mathcal{O}|=|\Gamma_{0}|/|\mathrm{stab}_{\Gamma_{0}}(\mathcal{O})|=2^{N}.$

## References

  * [BCG+09] J. Brodzki, S. J. Campbell, E. Guentner, G. A. Niblo, and N. J. Wright, _Property A and $\rm CAT(0)$ cube complexes_, J. Funct. Anal. 256 (2009), no. 5, 1408–1431. MR 2490224
  * [BFH04] Mladen Bestvina, Mark Feighn, and Michael Handel, _Solvable subgroups of ${\rm Out}(F_{n})$ are virtually Abelian_, Geom. Dedicata 104 (2004), 71–96. MR 2043955
  * [CFI16] Indira Chatterji, Talia Fernós, and Alessandra Iozzi, _The median class and superrigidity of actions on $\rm CAT(0)$ cube complexes_, J. Topol. 9 (2016), no. 2, 349–400, With an appendix by Pierre-Emmanuel Caprace. MR 3509968
  * [CN05] Indira Chatterji and Graham Niblo, _From wall spaces to $\rm CAT(0)$ cube complexes_, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 875–885. MR 2197811
  * [CS11] Pierre-Emmanuel Caprace and Michah Sageev, _Rank rigidity for CAT(0) cube complexes_ , Geom. Funct. Anal. 21 (2011), no. 4, 851–891. MR 2827012
  * [Fer18] Talia Fernós, _The Furstenberg-Poisson boundary and ${\rm CAT}(0)$ cube complexes_, Ergodic Theory Dynam. Systems 38 (2018), no. 6, 2180–2223. MR 3833346
  * [FFH23] Talia Fernós, David Futer, and Mark Hagen, _Homotopy equivalent boundaries of cube complexes_ , 2023.
  * [Fio18] Elia Fioravanti, _The Tits alternative for finite rank median spaces_ , Enseign. Math. 64 (2018), no. 1-2, 89–126. MR 3959849
  * [FLM18] Talia Fernós, Jean Lécureux, and Frédéric Mathéus, _Random walks and boundaries of $\rm CAT(0)$ cubical complexes_, Comment. Math. Helv. 93 (2018), no. 2, 291–333. MR 3811753
  * [FLM21] Talia Fernós, Jean Lécureux, and Frédéric Mathéus, _Contact graphs, boundaries, and a central limit theorem for cat(0) cubical complexes_ , 2021.
  * [Gen21] Anthony Genevois, _Cyclic hyperbolicity in cat(0) cube complexes_ , 2021.
  * [GJN23] Radhika Gupta, Kasia Jankiewicz, and Thomas Ng, _Groups acting on CAT(0) cube complexes with uniform exponential growth_ , Algebr. Geom. Topol. 23 (2023), no. 1, 13–42. MR 4567998
  * [MP22] Alexandre Martin and Piotr Przytycki, _Acylindrical actions for two-dimensional Artin groups of hyperbolic type_ , Int. Math. Res. Not. IMRN (2022), no. 17, 13099–13127. MR 4475273
  * [MS04] N. Monod and Y. Shalom, _Cocycle superrigidity and bounded cohomology for negatively curved spaces_ , J. Differential Geom. 67 (2004), no. 3, 395–455. MR 2153026 (2006g:53051)
  * [Nic04] Bogdan Nica, _Cubulating spaces with walls_ , Algebr. Geom. Topol. 4 (2004), 297–309. MR 2059193
  * [OP22] Damian Osajda and Piotr Przytycki, _Tits alternative for 2-dimensional $\rm CAT(0)$-dimensional complexes_, Forum Math. Pi 10 (2022), Paper No. e25, 19. MR 4522694
  * [Rol16] Martin Roller, _Poc sets, median algebras and group actions_ , 2016.
  * [Sag95] Michah Sageev, _Ends of group pairs and non-positively curved cube complexes_ , Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR 1347406
  * [SW05] Michah Sageev and Daniel T. Wise, _The Tits alternative for ${\rm CAT}(0)$ cubical complexes_, Bull. London Math. Soc. 37 (2005), no. 5, 706–710. MR 2164832
  * [Tit72] J. Tits, _Free subgroups in linear groups_ , J. Algebra 20 (1972), 250–270. MR 286898