# Generating Sets and Algebraic Properties of Pure Mapping Class Groups of
Infinite Graphs

George Domat, Hannah Hoganson, and Sanghoon Kwak

###### Abstract.

We completely classify the locally finite, infinite graphs with pure mapping
class groups admitting a coarsely bounded generating set. We also study
algebraic properties of the pure mapping class group: We establish a
semidirect product decomposition, compute first integral cohomology, and
classify when they satisfy residual finiteness and the Tits alternative. These
results provide a framework and some initial steps towards quasi-isometric and
algebraic rigidity of these groups.

##  1\. Introduction

A recent surge of interest in mapping class groups of infinite-type surfaces
has prompted the emergence of a “big” analog of $\operatorname{Out}(F_{n})$ as
well. Algom-Kfir–Bestvina [AB21] propose that the appropriate analog is the
group of self proper homotopy equivalences up to proper homotopy of a locally
finite, infinite graph.

One main difficulty of studying these “big” groups is that the classical
approach of geometric group theory is not applicable. In particular, the
mapping class groups of infinite-type surfaces and those of locally finite,
infinite graphs are generally not finitely generated, and not even compactly
generated. Fortunately, they are still Polish groups (separable and completely
metrizable topological groups), to which Rosendal provides a generalized
geometric group theoretic approach. The role of a finite or compact generating
set is replaced with a _coarsely bounded_ (CB) generating set. For example, a
group that admits a coarsely bounded generating set has a well-defined quasi-
isometry type [Ros22, Theorem 1.2 and Proposition 2.72], and a coarsely
bounded group is quasi-isometric to a point. A group with a coarsely bounded
neighborhood around the identity is said to be _locally coarsely bounded_ ,
which is equivalent to having a well-defined _coarse equivalence_ type, and is
necessary to have a coarsely bounded generating set. Using this framework,
Mann–Rafi [MR23] gave a classification of (tame) surfaces whose mapping class
groups are coarsely bounded, locally coarsely bounded, and generated by a
coarsely bounded set. This established a first step toward studying the coarse
geometry of mapping class groups of infinite-type surfaces. Recently, Thomas
Hill [Hil23] gave a complete classification of surfaces that have _pure_
mapping class groups with the aforementioned coarse geometric properties,
without the tameness condition.

In the authors’ previous work [DHK23], we gave a complete classification of
graphs with coarsely bounded, and locally coarsely bounded, pure mapping class
groups, the subgroup of the mapping class group fixing the ends of the graph
pointwise. In this paper, we provide the complete classification of infinite
graphs with CB-generated pure mapping class groups, fulfilling our goal to
provide a foundation for studying the coarse geometry of these groups. In the
following statement, $E$ refers to the space of ends of the graph ${\Gamma}$
and $E_{\ell}$ is the subset of ends accumulated by loops.

######  Theorem A.

Let ${\Gamma}$ be a locally finite, infinite graph. Then its pure mapping
class group, $\operatorname{PMap}({\Gamma})$, is _CB generated_ if and only if
either ${\Gamma}$ is a tree, or satisfies both:

  1. 1.

${\Gamma}$ has finitely many ends accumulated by loops, and

  2. 2.

there is no accumulation point in $E\setminus E_{\ell}$.

######  Remark 1.1.

Alternatively, we have a constructive description:
$\operatorname{PMap}({\Gamma})$ is CB generated if and only if ${\Gamma}$ can
be written (not necessarily uniquely) as a finite wedge sum of the four graphs
from Figure 1.

Figure 1. Every graph with a CB-generated pure mapping class group can be
written as a finite wedge sum of these four graphs. From left to right these
are: a single loop, a single ray, a Loch Ness monster graph, and a Millipede
monster graph.

Table 1 illustrates the complete classification of graphs with CB, locally CB,
and CB-generated pure mapping class group. Observe the trend that
$\operatorname{PMap}({\Gamma})$ admits more complicated coarse geometric
properties when ${\Gamma}$ has more complicated geometry.

$t=$ the number of components of ${\Gamma}\setminus{\Gamma}_{c}$ with infinite
end spaces Finite rank $r$ Infinite rank with $|E_{\ell}|=:n$ $r=0$abc
$r\in[1,\infty)$ $n=1$ $n\in[2,\infty)$ $n=\infty$ $t=0$ CB CB if
${\Gamma}=\leavevmode\hbox to17.87pt{\vbox
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abcc.cCBabac c aCB-generateda not locally CB $t\in[1,\infty)$ CB ab locally
CB ab ablocally CBab ablocally CBab not locally CB $t=\infty$ abcN/A
abcdefgN/A not locally CB not locally CB not locally CB

Table 1. (A and [DHK23]). Classification of ${\Gamma}$ with CB, locally CB,
and CB-generated $\operatorname{PMap}({\Gamma})$. Here ${\Gamma}_{c}$ is the
smallest connected subgraph that contains all the immersed loops in
${\Gamma}$. According to the implication CB $\Rightarrow$ CB-generated
$\Rightarrow$ locally CB, each cell is filled with the most general property.
For example, a cell labeled by CB-generated means that for such ${\Gamma}$,
$\operatorname{PMap}({\Gamma})$ is CB-generated but not CB.

The main tool used to prove A is the following semidirect decomposition of the
pure mapping class group (reminiscent of [APV20, Corollary 4] in the surface
setting):

######  Theorem B.

Let ${\Gamma}$ be a locally finite graph. Let
$\alpha=\max\\{0,|E_{\ell}({\Gamma})|-1\\}$ for $|E_{\ell}({\Gamma})|<\infty$
and $\alpha=\aleph_{0}$ otherwise. Then we have the following short exact
sequence,

$1\longrightarrow\overline{\operatorname{PMap}_{c}({\Gamma})}\longrightarrow\operatorname{PMap}({\Gamma})\longrightarrow\mathbb{Z}^{\alpha}\longrightarrow
1$

which splits. In particular, we have
$\operatorname{PMap}({\Gamma})=\overline{\operatorname{PMap}_{c}({\Gamma})}\rtimes\mathbb{Z}^{\alpha}.$

Here, $\overline{\operatorname{PMap_{\it c}}({\Gamma})}$ is the closure of the
group of compactly supported mapping classes and $\mathbb{Z}^{\alpha}$ is
generated by commuting loop shifts.

As a corollary, we compute the rank of the first integral cohomology of
$\operatorname{PMap}({\Gamma})$. This allows us to see that the number of ends
acummuluted by loops of a graph ${\Gamma}$ is an algebraic invariant of
$\operatorname{PMap}({\Gamma})$.

######  Corollary C.

For every locally finite, infinite graph ${\Gamma}$,

$\operatorname{rk}\left(H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})\right)=\begin{cases}0&\text{if
$|E_{\ell}|\leq 1$},\\\ n-1&\text{if $2\leq|E_{\ell}|=n<\infty$},\\\
\aleph_{0}&\text{otherwise}.\end{cases}$

We also show that $\operatorname{PMap}({\Gamma})$ distinguishes graphs of
finite rank from graphs of infinite rank. Recall a group is _residually
finite_ if it can be embedded into a direct product of finite groups.

######  Theorem D.

$\operatorname{PMap}({\Gamma})$ is residually finite if and only if ${\Gamma}$
has finite rank.

A group satisfies the Tits Alternative if every subgroup is either virtually
solvable or contains a nonabelian free group. Interestingly, it is exactly the
graphs with $\operatorname{PMap}({\Gamma})$ residually finite that satisfy the
Tits Alternative.

######  Theorem E.

$\operatorname{PMap}({\Gamma})$ satisfies the Tits Alternative if and only if
${\Gamma}$ has finite rank.

These three results are steps towards determining when the isomorphism type of
$\operatorname{PMap}({\Gamma})$ determines the graph ${\Gamma}$, as in the
surface case [BDR20].

If ${\Gamma}$ is the infinite rank graph with a single end (the Loch Ness
monster graph) and ${\Gamma}^{\prime}$ is the wedge sum of ${\Gamma}$ with a
single ray, then the groups $\operatorname{PMap}({\Gamma})$ and
$\operatorname{PMap}({\Gamma}^{\prime})$ inject into
$\operatorname{Out}(F_{\infty})$ and $\operatorname{Aut}(F_{\infty})$,
respectively, by [AB21, Theorem 3.1 and Lemma 3.2]. Thus we immediately get
the following corollary. We note that one can instead prove this directly,
e.g. see [Use].

######  Corollary F.

For $F_{\infty}$, the free group on a countably infinite set,
$\operatorname{Aut}(F_{\infty})$ and $\operatorname{Out}(F_{\infty})$ are not
residually finite and do not satisfy the Tits alternative.

### Comparison with Surfaces

The statement of B is exactly the same as for pure mapping class groups of
surfaces, seen in Aramayona–Patel–Vlamis [APV20]. Although the proof we give
is similar in spirit as well, we have to make use of different tools. In
[APV20] the authors make use of the _homology of separating curves_ on a
surface and build an isomorphism between the first cohomology of the pure
mapping class group and this homology group. For graphs, we do not have any
curves to take advantage of. Instead we use partitions of the space of ends
accumulated by loops. In order to make this precise and give this an algebraic
structure we make use of the group of locally constant integral functions on
$E_{\ell}({\Gamma})$, i.e., the zeroth Čech cohomology of
$E_{\ell}({\Gamma})$, denoted as $\mathring{C}(E_{\ell}({\Gamma}))$. On a
surface, any separating simple closed curve determines a partition of the end
space. We can use this to show that the first cohomology groups of pure
mapping class groups of graphs and surfaces are often in fact _naturally_
isomorphic. This also gives a slightly alternate proof of the main results in
[APV20].

######  Corollary G.

Let $S$ be an infinite-type surface of genus at least one and ${\Gamma}$ a
locally finite, infinite graph. If $E_{g}(S)$ is homeomorphic to
$E_{\ell}({\Gamma})$, then both $H^{1}(\operatorname{PMap}(S);\mathbb{Z})$ and
$H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})$ are isomorphic to
$\mathring{C}(E_{g}(S))\cong\mathring{C}(E_{\ell}({\Gamma}))$.

### Rigidity Questions

The results above fit naturally into a general question about (pure) mapping
class groups of infinite graphs. Namely: “How much does the group
$\operatorname{PMap}({\Gamma})$ determine the graph ${\Gamma}$?” One can
obtain more concrete questions by considering certain types of rigidity. We
will focus on algebraic and quasi-isometric rigidity. In the finite-type
setting, mapping class groups of surfaces and $\operatorname{Out}(F_{n})$ are
known to exhibit strong rigidity properties. Various results starting with
Ivanov [Iva88] (see also [McC86, Har86, BLM83, Iva97]) establish strong forms
of algebraic rigidity and Behrstock–Kleiner–Minsky–Mosher [BKMM12] establish
quasi-isometric rigidity for $\operatorname{Map}(S)$. For
$\operatorname{Out}(F_{n})$ we also have strong forms of algebraic rigidity
from the work of Farb–Handel [FH07] building on [Khr90, BV00] (see also
[HW20]). Quasi-isometric rigidity for $\operatorname{Out}(F_{n})$ is still
unknown.

For infinite-type surfaces, the work of Bavard–Dowdall–Rafi [BDR20]
established a strong form of algebraic rigidity á la Ivanov (see also
[HHMV18]). The question of quasi-isometric rigidity is still open, but
Mann–Rafi [MR23] give a classification of which mapping class groups of tame
infinite-type surfaces have a well-defined quasi-isometry type and which of
those are trivial. This allows one to begin to distinguish between some of the
mapping class groups (see also [SC20]).

One can ask the same rigidity questions for infinite graphs. The picture
becomes less clear than in the surface case. In particular, trees fail to have
algebraic rigidity for the _pure_ mapping class group, as they all have
trivial pure mapping class group. Failure is also present for the full mapping
class group. Let $T$ be the regular trivalent tree and let $T^{\prime}$ be the
wedge sum of $T$ with a single ray. Note that $E(T)=\mathcal{C}$, a Cantor
set, and $E(T^{\prime})=\mathcal{C}\sqcup\\{*\\}$, a Cantor set together with
a single isolated point. Now we have that
$\operatorname{Map}(T)=\operatorname{Homeo}(\mathcal{C})$ and
$\operatorname{Map}(T^{\prime})=\operatorname{Homeo}(\mathcal{C}\sqcup\\{*\\})$.
However, these two groups are isomorphic, as any homeomorphism fixes the extra
end $*$ of $T^{\prime}$. There are even more complicated examples of this
failure of algebraic rigidity for mapping class groups of trees that come from
work on Boolean algebras by McKenzie [McK77] answering a rigidity conjecture
of Monk [Mon75].

The results in this paper allow one to ask several natural rigidity questions
for the pure mapping class groups of infinite graphs. We will restrict to some
nice classes of graphs in order to state concrete questions. All of the
following families of graphs are CB generated by A and hence have a well-
defined quasi-isometry type. Let ${\Gamma}_{n}$ denote the graph with exactly
$n$ ends, each of which is accumulated by loops.

######  Question 1.2.

Let $n,m\geq 2$. If $\operatorname{PMap}({\Gamma}_{n})$ is quasi-isometric to
$\operatorname{PMap}({\Gamma}_{m})$, then does $n=m$?

By C we do know that $\operatorname{PMap}({\Gamma}_{n})$ is algebraically
isomorphic to $\operatorname{PMap}({\Gamma}_{m})$ if and only if $n=m$. We can
also use the fact that $\operatorname{PMap}({\Gamma}_{1})$ is CB to see that
$\operatorname{PMap}({\Gamma}_{1})$ is not quasi-isometric to
$\operatorname{PMap}({\Gamma}_{n})$ for any $n\neq 1$. However, the general
question is still open. In the authors’ previous work [DHK23], we computed
asymptotic dimension for all of these groups. However, it is infinite for
$n>1$. Therefore, in order to answer this question one would need to study
and/or develop other “big” quasi-isometry invariants.

Instead of comparing the effect of changing the number of ends accumulated by
loops, one could ask the same question for rays. Namely, let ${\Gamma}_{n,r}$
denote the graph with $n$ ends accumulated by loops and $r$ rays. We start by
asking for distinguishing features of “no ray” versus “one ray.”

######  Question 1.3.

Is $\operatorname{PMap}({\Gamma}_{n,0})$ quasi-isometric to
$\operatorname{PMap}({\Gamma}_{n,1})$?

In fact, here we do not even know algebraic rigidity.

######  Question 1.4.

Is $\operatorname{PMap}({\Gamma}_{n,0})$ isomorphic to
$\operatorname{PMap}({\Gamma}_{n,1})$?

The other large family of graphs with CB-generated pure mapping class groups
are the finite-type ones. Let $\Omega_{n,r}$ denote the graph of finite rank
$n$ with $r<\infty$ rays attached. We know that no
$\operatorname{PMap}(\Omega_{n,r})$ is isomorphic to any
$\operatorname{PMap}({\Gamma}_{m})$ by using either residual finiteness D or
the Tits alternative E. We do not know if any of them are quasi-isometric.
Note that $\operatorname{PMap}(\Omega_{n,r})$ is always finitely generated,
but this does not preclude it from being quasi-isometric to an uncountable
group.

######  Question 1.5.

Is $\operatorname{PMap}(\Omega_{m,r})$ ever quasi-isometric to
$\operatorname{PMap}({\Gamma}_{n})$, for $m,r,n>1$?

### Outline

In Section 2, we give background on mapping class groups of infinite graphs,
examples of elements in the pure mapping class group, and coarse geometry of
groups. In Section 3, we prove our semidirect product decomposition, B. We
also prove C in Section 3.5. By exploiting the semidirect decomposition of
$\operatorname{PMap}({\Gamma})$, we prove the CB-generation classification, A,
in Section 4. In Section 5 and Section 6, we finish by proving the residual
finiteness characterization D and Tits alternative characterization E.

### Acknowledgments

Thank you to Mladen Bestvina for providing an idea of the proof for 3.14 and
suggestion to use the zeroth Čech cohomology to prove 3.18. We also thank
Priyam Patel for helpful conversations toward Section 5 and D, along with
answering questions regarding [PV18] and [APV20]. Also we thank Camille Horbez
for clarifying the proof of 6.6.

The first author was supported in part by the Fields Institute for Research in
Mathematical Sciences, NSF DMS–1745670, and NSF DMS–2303262. The second author
was supported by NSF DMS–2303365. The third author acknowledges the support
from the University of Utah Graduate Research Fellowship.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Mapping class groups of infinite graphs
    2. 2.2 Elements of $\operatorname{PMap}({\Gamma})$
      1. 2.2.1 Loop swaps
      2. 2.2.2 Word maps
      3. 2.2.3 Loop shifts
    3. 2.3 Coarse geometry of groups
  3. 3 Semidirect product structure and cohomology
    1. 3.1 The case $|E_{\ell}|\leq 1$
    2. 3.2 Flux maps
    3. 3.3 Flux zero maps
    4. 3.4 Space of flux maps
    5. 3.5 The rank of integral cohomology
    6. 3.6 Relation to surfaces
  4. 4 CB generation classification
  5. 5 Residual finiteness
    1. 5.1 Forgetful map
    2. 5.2 Finite rank if and only if residually finite
  6. 6 Tits alternative
    1. 6.1 Infinite rank: Fails to satisfy TA
    2. 6.2 Finite rank: Satisfies TA

##  2\. Preliminaries

###  2.1. Mapping class groups of infinite graphs

Let ${\Gamma}$ be a locally finite, infinite graph. Informally, an _end_ of a
graph is a way to travel to infinity in the graph. The space of ends (or, the
end space), denoted by $E({\Gamma})$, is defined as:

$E({\Gamma})=\varprojlim_{K\subset{\Gamma}}\pi_{0}({\Gamma}\setminus K),$

where $K$ runs over compact sets of ${\Gamma}$ in the inverse limit. Then each
element of $E({\Gamma})$ is called an end of ${\Gamma}$. An end $e$ of
${\Gamma}$ is said to be accumulated by loops if the sequence of complementary
components in ${\Gamma}$ corresponding to $e$ only consists of infinite rank
graphs. Colloquially, if one continues to see loops along the way to $e$. We
denote by $E_{\ell}({\Gamma})$ the set of ends of ${\Gamma}$ accumulated by
loops. Note $E_{\ell}({\Gamma})$ is a closed subset of $E({\Gamma})$, and
$E({\Gamma})$ can be realized as a closed subset of a Cantor set (hence so is
$E_{\ell}({\Gamma})$). We say that the characteristic triple of ${\Gamma}$ is
the triple $(r({\Gamma}),E({\Gamma}),E_{\ell}({\Gamma}))$, where
$r({\Gamma})\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ is the rank of
$\pi_{1}({\Gamma})$.

Now we define the mapping class group of a locally finite, infinite graph
${\Gamma}$. Recall that a map is proper if the pre-image of every compact set
is compact.

######  Definition 2.1.

[AB21] The mapping class group of ${\Gamma}$, denoted
$\operatorname{Map}({\Gamma})$, is the group of proper homotopy classes of
proper homotopy equivalences of ${\Gamma}$. The pure mapping class group,
denoted $\operatorname{PMap}({\Gamma})$, is the closed subgroup consisting of
maps that fix the ends of ${\Gamma}$ pointwise. More precisely, it is the
kernel of the action of $\operatorname{Map}({\Gamma})$ on the end space
$(E({\Gamma}),E_{\ell}({\Gamma}))$ by homeomorphisms, hence fitting into the
following short exact sequence:

$1\longrightarrow\operatorname{PMap}({\Gamma})\longrightarrow\operatorname{Map}({\Gamma})\longrightarrow\operatorname{Homeo}(E,E_{\ell})\longrightarrow
1$

When $E({\Gamma})\setminus E_{\ell}({\Gamma})$ is nonempty and compact, we can
further decompose $\operatorname{PMap}({\Gamma})$ into subgroups of _core
maps_ and of _ray maps_. To state the result, we need to introduce a few
concepts.

######  Definition 2.2.

Let ${\Gamma}$ be a locally finite, infinite graph. Denote by ${\Gamma}_{c}$
the core graph of ${\Gamma}$, the smallest connected subgraph of ${\Gamma}$
that contains all immersed loops in ${\Gamma}$. When $E({\Gamma})\setminus
E_{\ell}({\Gamma})$ is nonempty, pick $e_{0}\in E({\Gamma})\setminus
E_{\ell}({\Gamma})$ and denote by ${\Gamma}_{c}^{*}$ be the subgraph
consisting of ${\Gamma}_{c}$ and a choice of embedded ray in ${\Gamma}$
limiting to $e_{0}$ such that the ray intersects ${\Gamma}_{c}$ in exactly one
point.

Define $\pi_{1}({\Gamma}_{c}^{*},e_{0})$ to be the set of proper homotopy
equivalence classes of lines in ${\Gamma}_{c}^{*}$, both ends of which limit
to $e_{0}$. We endow it with a group structure by concatenation. This group is
naturally isomorphic to $\pi_{1}({\Gamma}_{c}^{*},p)$ for any choice of
basepoint $p\in{\Gamma}_{c}^{*}$. Finally, define $\mathcal{R}$ as the group
of maps $h:E({\Gamma})\to\pi_{1}({\Gamma}_{c}^{*},e_{0})$ such that

  1. (i)

$h(e_{0})=1$, and

  2. (ii)

$h$ is locally constant,

where the group operation is the pointwise multiplication in
$\pi_{1}({\Gamma}_{c}^{*},e_{0})$.

We have the following decomposition of $\operatorname{PMap}({\Gamma})$:

######  Proposition 2.3 ([AB21, Corollary 3.9]).

Let ${\Gamma}$ be a locally finite, infinite graph with $E({\Gamma})\setminus
E_{\ell}({\Gamma})$ nonempty and compact. Then

$\operatorname{PMap}({\Gamma})\cong\mathcal{R}\rtimes\operatorname{PMap}({\Gamma}_{c}^{*}).$

In particular, when ${\Gamma}$ has finite rank $n\geq 0$ and finitely many
ends, say $|E({\Gamma})|=e$, then

$\operatorname{PMap}({\Gamma})\cong\begin{cases}\operatorname{Out}(F_{n}),&\text{if
$e=0$,}\\\ F_{n}^{e-1}\rtimes\operatorname{Aut}(F_{n}),&\text{if $e\geq
1$.}\end{cases}$

######  Remark 2.4.

Any time $K$ is a connected, compact subgraph of a locally finite, infinite
graph ${\Gamma}$, we use $\operatorname{PMap}(K)$ to refer to the group of
proper homotopy equivalences of $K$ that fix $\partial K$ pointwise up to
proper homotopy fixing $\partial K$. This group is naturally isomorphic to the
group $\operatorname{PMap}(\tilde{K})$ where $\tilde{K}$ is the graph $K$
together with a ray glued to each point in $\partial K$. Applying the above
proposition we see that $\operatorname{PMap}(K)$ is always of the form
$F_{n}^{e-1}\rtimes\operatorname{Aut}(F_{n})$ for some $n$ and $e$ because $K$
is always a proper subset of ${\Gamma}$, so $\partial K$ is nonempty.

The pure mapping class group $\operatorname{PMap}({\Gamma})$ records the
internal symmetries of ${\Gamma}$. Contractible graphs (trees) have no
internal symmetries. This follows from the work of
Ayala–Dominguez–Márquez–Quintero [ADMQ90]. They give a proper homotopy
equivalence classification of locally finite, infinite graphs.

######  Theorem 2.5 ([ADMQ90, Theorem 2.7]).

Let ${\Gamma}$ and ${\Gamma}^{\prime}$ be two locally finite graphs of the
same rank. A homeomorphism of end spaces
$(E({\Gamma}),E_{\ell}({\Gamma}))\to(E({\Gamma}^{\prime}),E_{\ell}({\Gamma}^{\prime}))$
extends to a proper homotopy equivalence ${\Gamma}\to{\Gamma}^{\prime}$. If
${\Gamma}$ and ${\Gamma}^{\prime}$ are trees, then this extension is unique up
to proper homotopy.

The second statement of 2.5 implies the following.

######  Proposition 2.6.

Let ${\Gamma}$ be a locally finite, infinite graph with $\pi_{1}({\Gamma})=1$.
Then $\operatorname{PMap}({\Gamma})=1$.

In [AB21] the authors endow $\operatorname{Map}({\Gamma})$ with the compact-
open topology and show that this gives $\operatorname{Map}({\Gamma})$, and
hence $\operatorname{PMap}({\Gamma})$, the structure of a Polish group. A
neighborhood basis about the identity for the topology is given by sets of the
form

$\displaystyle\mathcal{V}_{K}$
$\displaystyle:=\\{[f]\in\operatorname{Map}({\Gamma})|~{}\exists
f^{\prime}\in[f]\text{ s.t. }f^{\prime}|_{K}=\operatorname{id}\text{ and }$
$\displaystyle f^{\prime}\text{ preserves the complementary components of $K$
setwise}\\}$

where $K$ is a compact subset of ${\Gamma}$.

Recall the support of a continuous map $\phi:X\to X$ is the closure of the set
of points $x\in X$ such that $\phi(x)\neq x$. The group of compactly supported
mapping classes, denoted by $\operatorname{PMap_{\it c}}({\Gamma})$, is the
subgroup of $\operatorname{PMap}({\Gamma})$ consisting of classes that have a
compactly supported representative. Its closure in this topology is denoted by
$\overline{\operatorname{PMap_{\it c}}({\Gamma})}$ and it is a closed (hence
Polish) subgroup of $\operatorname{PMap}({\Gamma})$.

As proper homotopy equivalences are not necessarily injective, unlike
homeomorphisms, we need the following alternate notion of support for a proper
homotopy equivalence.

######  Definition 2.7.

We say that $[f]\in\operatorname{Map}({\Gamma})$ is totally supported on
$K\subset{\Gamma}$ if there is a representative $f^{\prime}\in[f]$ so that
$f^{\prime}(K)=K$ and $f^{\prime}|_{{\Gamma}\setminus K}=\operatorname{id}$.

To see how a proper homotopy equivalence can have different support and total
support, consider a rose graph with two loops labeled by $a_{1}$ and $a_{2}$.
Then a (proper) homotopy equivalence mapping $a_{1}$ to $a_{1}a_{2}$ which is
the identity elsewhere is supported on $a_{1}$, but not totally supported on
$a_{1}$. It is totally supported on $a_{1}\cup a_{2}$. This is in contrast
with homeomorphisms on surfaces, where $f$ is supported on $K$ if and only if
$f$ is totally supported on $K$.

As mapping class groups of graphs are independent of the proper homotopy
equivalence representative of the graph, it is often useful to consider a
‘standard’ representative within a proper homotopy equivalence class of
graphs.

######  Definition 2.8.

A locally finite graph, ${\Gamma}$, is in standard form if ${\Gamma}$ is a
tree with loops attached at some of the vertices. We endow ${\Gamma}$ with the
path metric that assigns each edge length $1$.

We also give special names to specific graphs that we will reference often.

######  Definition 2.9.

The Loch Ness Monster graph is the graph with characteristic triple $(\infty,\
\\{*\\},\ \\{*\\})$. The Millipede Monster graph is the graph with
characteristic triple $(\infty,\ \\{0\\}\cup\\{\frac{1}{n}\ |\
n\in\mathbb{Z}_{>0}\\},\ \\{0\\})$. A monster graph refers to either one of
these.

###  2.2. Elements of $\operatorname{PMap}({\Gamma})$

Here we give a brief treatment of elements of $\operatorname{PMap}({\Gamma})$.
For more detailed definitions with examples, see [DHK23, Section 3].

####  2.2.1. Loop swaps

A Loop swap is an order 2 proper homotopy equivalence induced from a
transposition automorphism of a free group. It is totally supported on a
compact set. More precisely, we define it as follows.

######  Definition 2.10.

Let ${\Gamma}$ be a locally finite graph in standard form with
$\operatorname{rk}{\Gamma}\geq 2$. Let $A$ and $B$ be disjoint finite subsets
of loops such that $|A|=|B|$. Then the loop swap $\mathcal{L}(A,B)$ is a
proper homotopy equivalence induced from the group isomorphism on
$\pi_{1}({\Gamma})$ swapping the free factors corresponding to $A$ and $B$.

More concretely, pick a basepoint $p\in{\Gamma}$ and collapse each maximal
tree of the subgraphs corresponding to $A$ and $B$ in $\pi_{1}({\Gamma},p)$.
This results in two roses of $|A|=|B|$ petals. Then swap the two roses,
followed by blowing up each rose to the original subgraph. Define
$\mathcal{L}(A,B)$ as the composition of these three maps. Note
$\mathcal{L}(A,B)\in\operatorname{PMap_{\it c}}({\Gamma}).$

As mentioned above, loop swaps of a graph correspond to the transposition free
group automorphisms, which are part of generating set for
$\operatorname{Aut}(F_{n})$. (See Section 4).

####  2.2.2. Word maps

Next, word maps, which are the most diverse kind of mapping classes among the
three kinds of maps introduced in this section.

######  Definition 2.11.

Let ${\Gamma}$ be a locally finite graph with $\operatorname{rk}{\Gamma}\geq
1$, with a base point $p\in{\Gamma}$. Let $w\in\pi_{1}({\Gamma},p)$ and $I$ be
an interval in an edge of ${\Gamma}$. Then the word map $\varphi_{(w,I)}$ is a
proper homotopy equivalence that maps $I$ to a path in ${\Gamma}$ determined
by $w\in\pi_{1}({\Gamma},p)$ and is the identity outside of $I$.

See [DHK23, Section 3.3] for a careful construction of these maps. Note
$\varphi_{(w,I)}$ is supported on $I$, but in general not _totally_ supported
on $I$. Rather, it is totally supported on the compact set that is the union
of $I$ with a path in ${\Gamma}$ determined by $w\in\pi_{1}({\Gamma},p)$.

The following two properties of word maps will be important in Section 4.

######  Lemma 2.12 ([DHK23, Lemma 3.5]).

If $I$ is contained in an edge of ${\Gamma}\setminus{\Gamma}_{c}$ and
$w_{1},w_{2}$ are elements in $\pi_{1}({\Gamma},p)$, then

$[\varphi_{(w_{1},I)}\circ\varphi_{(w_{2},I)}]=[\varphi_{(w_{1}w_{2},I)}]$

in $\operatorname{PMap}({\Gamma})$.

######  Lemma 2.13 ([DHK23, Lemma 3.10]).

Let $I$ be an interval of ${\Gamma}$ which is outside of ${\Gamma}_{c}$, and
$\psi\in\operatorname{PMap}({\Gamma})$ be totally supported on a compact
subgraph of ${\Gamma}_{c}$. Then

$\psi\circ[\varphi_{(w,I)}]\circ\psi^{-1}=[\varphi_{(\psi_{*}(w),I)}].$

In particular, we can use 2.13 when $\psi$ is a loop swap.

####  2.2.3. Loop shifts

Loop shifts are to graphs as handle shifts are to surfaces, which were
introduced in Patel–Vlamis [PV18]. We first define a loop shift on the
standard form of the graph $\Lambda$, the graph with characteristic triple
$(\infty,\\{e_{-},e_{+}\\},\\{e_{-},e_{+}\\})$. (See 2.8.) Embed $\Lambda$ in
$\mathbb{R}^{2}$ by identifying the maximal tree with the $x$-axis such that
$e_{\pm}$ is identified with $\pm\infty$ of the $x$-axis, and each vertex is
identified with an integer point in the $x$-axis. Identify the loops with the
circles $\\{(x-n)^{2}+(y-\frac{1}{4})^{2}=\frac{1}{16}\\}_{n\in\mathbb{Z}}$.
Note these circles are tangent to the integer points
$\\{(n,0)\\}_{n\in\mathbb{Z}}$, thus representing the loops in $\Lambda$. Now
define the primitive loop shift $h$ on $\Lambda$ as the horizontal translation
$x\mapsto x+1$. One can also omit some loops from $\Lambda$ and define the
loop shift to avoid those loops. For a more general definition, see [DHK23,
Section 3.4].

######  Definition 2.14.

Now we define the loop shift on a locally finite, infinite graph ${\Gamma}$
with $|E_{\ell}|\geq 2$. Pick two distinct ends $e_{-},e_{+}\in
E_{\ell}({\Gamma})$ accumulated by loops. By considering a standard form of
${\Gamma}$, we can find an embedded ladder graph $\Lambda$ in ${\Gamma}$ such
that $e_{\pm}$ is identified with $e_{\pm}$ of $\Lambda$, respectively. Now
define the primitive loop shift $h$ on ${\Gamma}$ associated to
$(e_{-},e_{+})$ as the proper homotopy equivalence induced from the primitive
loop shift on the embedded ladder graph $\Lambda$. For the rest of the graph,
define $h$ to be the identity outside of the $\frac{1}{2}$-neighborhood of
$\Lambda$ and interpolate between the shift and the identity on the
$\frac{1}{2}$-neighborhood.

Finally, a proper homotopy equivalence $f$ on ${\Gamma}$ is a loop shift if
$f=h^{n}$ for some primitive loop shift $h$ and
$n\in\mathbb{Z}\setminus\\{0\\}$.

###  2.3. Coarse geometry of groups

######  Definition 2.15.

Let $A$ be a subset of a topological group $G$. Then $A$ is coarsely bounded
(CB) in $G$ if for every continuous isometric action of $G$ on a metric space,
every orbit is bounded.

We say a group is CB-generated if it has an algebraic generating set that is
CB. Similarly, a group is locally CB if it admits a CB neighborhood of the
identity. In Section 4, we will construct a CB-generating set for the pure
mapping class groups of certain graphs, proving the if direction of A. On the
other hand, we have previously classified which graphs have CB or locally CB
mapping class groups:

######  Theorem 2.16 ([DHK23, Theorem A, D]).

Let ${\Gamma}$ be a locally finite, infinite graph. Then its pure mapping
class group $\operatorname{PMap}({\Gamma})$ is _coarsely bounded_ if and only
if one of the following holds:

  * •

${\Gamma}$ has rank zero, or

  * •

${\Gamma}$ has rank one, and has one end, or

  * •

${\Gamma}$ is a monster graph with finitely many rays attached.

Moreover, $\operatorname{PMap}({\Gamma})$ is _locally coarsely bounded_ if and
only if one of the following holds:

  * •

${\Gamma}$ has finite rank, or

  * •

${\Gamma}$ satisfies both:

    1. 1.

$|E_{\ell}({\Gamma})|<\infty$, and

    2. 2.

only finitely many components of ${\Gamma}\setminus{\Gamma}_{c}$ have infinite
end spaces.

######  Remark 2.17.

Mirroring the constructive description in 1.1 of the CB-generated
$\operatorname{PMap}({\Gamma})$ classification, we can alternatively
characterize the locally CB condition as: $\operatorname{PMap}({\Gamma})$ is
locally CB if and only if ${\Gamma}$ can be written as a finite wedge sum of
single loops, monster graphs, and trees.

After confirming that a group is CB-generated, the Rosendal framework enables
the exploration of the group through the lens of coarse geometry.

######  Theorem 2.18.

[Ros22, Theorem 1.2, Proposition 2.72] Let $G$ be a CB-generated Polish group.
Then $G$ has a well-defined quasi-isometry type. Namely, any two CB-generating
sets for $G$ give rise to quasi-isometric word metrics on $G$.

##  3\. Semidirect product structure and cohomology

In this section, we prove:

######  Theorem 3.1 (B, revisited).

Let ${\Gamma}$ be a locally finite graph. Let
$\alpha=\max\\{0,|E_{\ell}({\Gamma})|-1\\}$ for $|E_{\ell}({\Gamma})|<\infty$
and $\alpha=\aleph_{0}$ otherwise. Then we have the following short exact
sequence,

$1\longrightarrow\overline{\operatorname{PMap}_{c}({\Gamma})}\longrightarrow\operatorname{PMap}({\Gamma})\longrightarrow\mathbb{Z}^{\alpha}\longrightarrow
1$

which splits. In particular, we have
$\operatorname{PMap}({\Gamma})=\overline{\operatorname{PMap}_{c}({\Gamma})}\rtimes\mathbb{Z}^{\alpha}.$

The map to $\mathbb{Z}^{\alpha}$ is defined using _flux maps_ , which were
first defined for locally finite, infinite graphs in [DHK23]. We quickly treat
the case when the graph has at most one end accumulated by loops in Section
3.1. Then in Section 3.2, we recap the necessary definitions for flux maps and
further expand on their properties. In Section 3.3, we characterize
$\overline{\operatorname{PMap}_{c}({\Gamma})}$ as the common kernel of all
flux maps (3.11), which provides the left side of the desired splitting short
exact sequence. Then in Section 3.4, we construct the other side of the short
exact sequence by finding a section, proving B. This requires us to study the
space of flux maps, which is done in the same subsection. As an application,
in Section 3.5 we compute the first integral cohomology of
$\operatorname{PMap}({\Gamma})$. Finally, we show the same approach could have
been applied to infinite-type surfaces in Section 3.6 to recover the surface
version of B by Aramayona–Patel–Vlamis [APV20], by showing that there is a
natural isomorphism between the first cohomology of the pure mapping class
groups of infinite-type surfaces and infinite graphs.

###  3.1. The case $|E_{\ell}|\leq 1$

######  Proposition 3.2.

Let ${\Gamma}$ be a locally finite, infinite graph with $|E_{\ell}|\leq 1$.
Then
$\operatorname{PMap}({\Gamma})=\overline{\operatorname{PMap}_{c}({\Gamma})}$.
Furthermore, if $\lvert E_{\ell}\rvert=0$, then
$\operatorname{PMap}({\Gamma})=\operatorname{PMap}_{c}({\Gamma})$.

###### Proof.

The case when $|E_{\ell}({\Gamma})|=1$ is the result of [DHK23, Corollary
4.5]. Now we assume $|E_{\ell}({\Gamma})|=0$, i.e., ${\Gamma}$ has finite
rank.

Let $f\in\operatorname{PMap}({\Gamma})$. Because $f$ is proper,
$f^{-1}({\Gamma}_{c})$ is compact. Thus, there is some connected compact set
$K$ such that ${\Gamma}_{c}\cup f^{-1}({\Gamma}_{c})\subset K$. Now
$f|_{{\Gamma}\setminus K}$ is a proper homotopy equivalence between two
contractible sets and thus $f$ can be homotoped to be totally supported on
$K$. Hence, we conclude $f\in\operatorname{PMap}_{c}({\Gamma})$. ∎

###  3.2. Flux maps

We begin the case when $|E_{\ell}|\geq 2$, where the flux maps come onto the
scene. Here we recap the definitions and properties of flux maps developed in
[DHK23, Section 7].

Let ${\Gamma}$ be a locally finite, infinite graph with $|E_{\ell}|\geq 2$.
For each nonempty, proper, clopen subset $\mathcal{E}$ of $E_{\ell}$, we will
construct a flux map $\Phi_{\mathcal{E}}$, which will evaluate to 1 for every
primitive loop shift that goes from an end in $E_{\ell}\setminus\mathcal{E}$
to an end in $\mathcal{E}$. We fix such a subset $\mathcal{E}$ for this
discussion.

After potentially applying a proper homotopy equivalence, we can put
${\Gamma}$ into a standard form so that there is a maximal tree $T$ and a
choice of $x_{0}$ in $T$ such that ${\Gamma}\setminus\\{x_{0}\\}$ defines a
partition of the ends that is compatible with the partition
$\mathcal{E}\sqcup(E_{\ell}\setminus\mathcal{E})$ of $E_{\ell}$. That is, the
components of ${\Gamma}\setminus\\{x_{0}\\}$ determine a partition
$E=\bigsqcup_{i=1}^{m}\mathcal{F}_{i}$ so that we can rewrite as
$\mathcal{E}=\bigsqcup_{i=1}^{k}(\mathcal{F}_{i}\cap E_{\ell})$ and
$E_{\ell}\setminus\mathcal{E}=\bigsqcup_{i=k+1}^{m}(\mathcal{F}_{i}\cap
E_{\ell})$.

Now we group the components of ${\Gamma}\setminus\\{x_{0}\\}$ by the set
$\mathcal{E}$. Let ${\Gamma}_{+}$ and ${\Gamma}_{-}$ be the unions of the
closures of the components of ${\Gamma}\setminus\\{x_{0}\\}$ so that
$E_{\ell}({\Gamma}_{+})=\mathcal{E}$ and
$E_{\ell}({\Gamma}_{-})=E_{\ell}\setminus\mathcal{E}$. More precisely,
${\Gamma}_{+}$ is exactly the union of the complementary components of $x_{0}$
with end spaces corresponding to $\mathcal{F}_{1},\ldots,\mathcal{F}_{k}$
together with adding back in $x_{0}$. Similarly, ${\Gamma}_{-}$ is the union
of the components corresponding to $\mathcal{F}_{k+1},\ldots,\mathcal{F}_{m}$,
together with $x_{0}$. Finally, let $T_{-}$ be the maximal tree of
$\Gamma_{-}$ contained in $T$. Define for each $n\in\mathbb{Z}$:

$\displaystyle\Gamma_{n}$
$\displaystyle:=\begin{cases}\overline{\Gamma_{-}\cup B_{n}(x_{0})}&\text{ if
}n\geq 0,\\\ \left(\Gamma_{-}\setminus B_{n}(x_{0})\right)\cup T_{-}&\text{ if
}n<0,\end{cases}$

where $B_{n}(x_{0})$ is the open metric ball of radius $n$ about $x_{0}$. See
[DHK23, Section 7.2] for more details and pictures of the ${\Gamma}_{n}$’s.

Recall that a subgroup $A$ of a group $G$ is a free factor if there exists
another subgroup $P$ such that $G=A*P$. Given a free factor $A$ of $B$, we
define the corank of $A$ in $B$, denoted by $\operatorname{cork}(B,A)$, as the
rank of $B/\langle\\!\langle A\rangle\\!\rangle$, the quotient of $B$ by the
normal closure of $A$. For the ${\Gamma}_{n}$ defined above we write
$A_{n}=\pi_{1}({\Gamma}_{n},x_{0})$, the free factor determined by the
subgraph ${\Gamma}_{n}$.

Denote by $\operatorname{PPHE}({\Gamma})$ the group of proper homotopy
equivalences on ${\Gamma}$ that fix the ends of ${\Gamma}$ pointwise and fix
the basepoint $x_{0}$, i.e., the group of _pure_ proper homotopy equivalences.
Any pure mapping class can be properly homotoped to fix a point, hence every
pure mapping class has a representative in $\operatorname{PPHE}({\Gamma})$.
Note a proper homotopy equivalence on ${\Gamma}$ induces an isomorphism on the
level of fundamental group. Hence, with our choice of basepoint
$x_{0}\in{\Gamma}$, for each element $f\in\operatorname{PPHE}({\Gamma})$, we
denote by $f_{*}$ the induced map on $\pi_{1}({\Gamma},x_{0})$.

######  Definition 3.3 ([DHK23, Definition 7.9]).

Given $f\in\operatorname{PPHE}(\Gamma)$, we say that a pair of integers,
$(m,n)$, with $m>n$, is admissible for $f$ if

  1. (1)

$A_{n}$ and $f_{*}(A_{n})$ are free factors of $A_{m}$, and

  2. (2)

both $\operatorname{cork}(A_{m},A_{n})$ and
$\operatorname{cork}(A_{m},f_{*}(A_{n}))$ are finite.

In [DHK23, Corollary 7.8], we showed that for every
$f\in\operatorname{PPHE}({\Gamma})$, and $n\in\mathbb{Z}$, there exist
$m\in\mathbb{Z}$ such that $m>n$ and $(m,n)$ is admissible for $f$. Hence, we
can define:

######  Definition 3.4.

For a map $f\in\operatorname{PPHE}(\Gamma)$ and an admissible pair $(m,n)$ for
$f$, we let

$\displaystyle\phi_{m,n}(f):=\operatorname{cork}(A_{m},A_{n})-\operatorname{cork}(A_{m},f_{*}(A_{n})).$

Call such a $\phi_{m,n}$ a PPHE-flux map.

######  Lemma 3.5 ([DHK23, Lemma 7.10]).

The PPHE-flux of a map $f\in\operatorname{PPHE}({\Gamma})$ is well-defined
over the choice of admissible pair $(m,n)$. That is, if $(m,n)$ and
$(m^{\prime},n^{\prime})$ are two admissible pairs for the map
$f\in\operatorname{PPHE}(\Gamma)$ then
$\phi_{m,n}(f)=\phi_{m^{\prime},n^{\prime}}(f)$.

Furthermore:

######  Proposition 3.6 ([DHK23, Proposition 7.11 and Lemma 7.12]).

The PPHE-flux maps are homomorphisms. Moreover, for any nonempty proper clopen
subset $\mathcal{E}$ of $E_{\ell}$, if $f,g\in\operatorname{PPHE}({\Gamma})$
are properly homotopic, then $\phi_{\mathcal{E}}(f)=\phi_{\mathcal{E}}(g)$.

Hence, the PPHE-flux map factors through $\operatorname{PMap}({\Gamma})$, so
we can define the flux map on $\operatorname{PMap}({\Gamma})$ as follows.

######  Definition 3.7.

For each nonempty proper clopen subset $\mathcal{E}$ of $E_{\ell}$, we define
the flux map as:

$\displaystyle\Phi_{\mathcal{E}}:\operatorname{PMap}(\Gamma)\rightarrow\mathbb{Z}$
$\displaystyle[f]\mapsto\phi_{\mathcal{E}}(f),$

which is a well-defined homomorphism by 3.6.

This independence of the choice of admissible pairs further implies the
independence of the choice of the basepoint $x_{0}$.

######  Lemma 3.8 (Independence to choice of $x_{0}$).

For a nonempty proper clopen subset $\mathcal{E}$ of $E_{\ell}$, let $x_{0}$
and $x_{0}^{\prime}$ be two different points that realize the partition
$E_{\ell}=\mathcal{E}\sqcup(E_{\ell}\setminus\mathcal{E})$. Say
$\phi_{\mathcal{E}}$ and $\phi_{\mathcal{E}}^{\prime}$ are the flux maps
constructed from $x_{0}$ and $x_{0}^{\prime}$ respectively, with the same
orientation;
$E_{\ell}({\Gamma}_{+})=E_{\ell}({\Gamma}_{+}^{\prime})=\mathcal{E}$. Then
$\phi_{\mathcal{E}}=\phi_{\mathcal{E}}^{\prime}$.

###### Proof.

Note $x_{0}$ and $x_{0}^{\prime}$ together cut ${\Gamma}$ into three parts
(not necessarily connected), where two of them are of infinite rank and
realize $\mathcal{E}$ and $E_{\ell}\setminus\mathcal{E}$ respectively, and the
middle part is of finite rank (but not necessarily compact), and we call it
$M$.

Let $\\{{\Gamma}_{n}\\}$ and $\\{{\Gamma}_{n}^{\prime}\\}$ be the chains of
graphs used to define $\phi_{\mathcal{E}}$ and $\phi_{\mathcal{E}}^{\prime}$
respectively. Then since $\phi_{\mathcal{E}}$ and
$\phi_{\mathcal{E}^{\prime}}$ are in the same direction, there exists
$k\in\mathbb{Z}$ such that $A_{n+k}=A_{n}^{\prime}$ for all $n\in\mathbb{Z}$.
To be precise, this holds for $k$ such that ${\Gamma}_{k}$ and
${\Gamma}_{0}^{\prime}$ have the same core graph. Now, given
$f\in\operatorname{PMap}({\Gamma})$ and an admissible pair $(m,n)$ for $f$ at
$x_{0}$, the pair $(m-k,n-k)$ is admissible for $f$ at $x_{0}^{\prime}$. Then

$\displaystyle(\phi_{\mathcal{E}})_{m,n}(f)$
$\displaystyle=\operatorname{cork}(A_{m},A_{n})-\operatorname{cork}(A_{m},f_{*}(A_{n}))$
$\displaystyle=\operatorname{cork}(A^{\prime}_{m-k},A^{\prime}_{n-k})-\operatorname{cork}(A^{\prime}_{m-k},f_{*}(A^{\prime}_{n-k}))=(\phi_{\mathcal{E}}^{\prime})_{m-k,n-k}(f).$

All in all, the independence of the choice of admissible pairs by 3.5 proves
that $\phi_{{\mathcal{E}}}(f)=\phi_{{\mathcal{E}}}^{\prime}(f)$. Since $f$ was
chosen arbitrarily, this concludes the proof. ∎

Therefore, for each nonempty proper clopen subset $\mathcal{E}$ of $E_{\ell}$,
we can write the resulting flux map as $\phi_{\mathcal{E}}$ without specifying
$x_{0}$.

We end this subsection by exploring basic properties of flux maps, to be used
in subsequent subsections. Note that flux maps inherit the group operation
from $\operatorname{Hom}(\operatorname{PMap}({\Gamma}),\mathbb{Z})$; pointwise
addition.

######  Proposition 3.9.

Let $\mathcal{E}\subset E_{\ell}$ be a nonempty proper clopen subset of
$E_{\ell}$, where $|E_{\ell}|\geq 2$. Let $A,B$ and $B^{\prime}$ be nonempty
proper clopen subsets of $E_{\ell}$, such that $A$ and $B$ are disjoint, and
$B$ is a proper subset of $B^{\prime}$. Then the following hold:

  1. (i)

$\Phi_{\mathcal{E}^{c}}=-\Phi_{\mathcal{E}}$.

  2. (ii)

$\Phi_{A\sqcup B}=\Phi_{A}+\Phi_{B}.$

  3. (iii)

$\Phi_{B^{\prime}\setminus B}=\Phi_{B^{\prime}}-\Phi_{B}$.

###### Proof.

We first note that (iii) follows from (ii), noting that $B^{\prime}\setminus
B$ and $B$ are disjoint. Hence, it suffices to prove (i) and (ii).

  1. (i)

Let $f\in\operatorname{PPHE}({\Gamma})$ and $\mathcal{E}\subset E_{\ell}$ be a
nonempty proper clopen subset. Choose $g\in\operatorname{PPHE}({\Gamma})$ to
be a proper homotopy inverse of $f$. Take ${\Gamma}_{L}$ and ${\Gamma}_{R}$
with ${\Gamma}_{L}\subset{\Gamma}_{R}$ to be an admissible pair of graphs for
$f$ and $g$ with respect to $\mathcal{E}.$ Fixing ${\Gamma}_{L}$, we can
enlarge ${\Gamma}_{R}$ so that
$({\Gamma}\setminus{\Gamma}_{L},{\Gamma}\setminus{\Gamma}_{R})$ is an
admissible pair for $f$ with respect to $\mathcal{E}^{c}$. Note
$({\Gamma}_{R},{\Gamma}_{L})$ is still an admissible pair of graphs for $f$
with respect to $\mathcal{E}$. In summary, we have:

     * •

$f({\Gamma}_{L})\subset{\Gamma}_{R},\quad g({\Gamma}_{L})\subset{\Gamma}_{R}$

     * •

$f({\Gamma}\setminus{\Gamma}_{R})\subset{\Gamma}\setminus{\Gamma}_{L}$,

     * •

$\operatorname{cork}(\pi_{1}({\Gamma}_{R}),\pi_{1}({\Gamma}_{L}))<\infty,\quad\operatorname{cork}(\pi_{1}({\Gamma}_{R}),f_{*}(\pi_{1}({\Gamma}_{L})))<\infty$.

     * •

$\operatorname{cork}(\pi_{1}({\Gamma}_{R}),g_{*}(\pi_{1}({\Gamma}_{L})))<\infty$.

     * •

$\operatorname{cork}(\pi_{1}({\Gamma}\setminus{\Gamma}_{L}),\pi_{1}({\Gamma}\setminus{\Gamma}_{R}))<\infty,\quad\operatorname{cork}(\pi_{1}({\Gamma}\setminus{\Gamma}_{L}),f_{*}(\pi_{1}({\Gamma}\setminus{\Gamma}_{R})))<\infty$.

Because $f_{*}$ is a $\pi_{1}$-isomorphism, we have the following three
different free factor decompositions of $\pi_{1}({\Gamma})$:

$\displaystyle\pi_{1}({\Gamma})$
$\displaystyle=f_{*}(\pi_{1}({\Gamma}_{R}))\ast
f_{*}(\pi_{1}({\Gamma}\setminus{\Gamma}_{R})),$
$\displaystyle\pi_{1}({\Gamma})$
$\displaystyle=\pi_{1}({\Gamma}_{R})\ast\pi_{1}({\Gamma}\setminus{\Gamma}_{R}),\text{
and}$ $\displaystyle\pi_{1}({\Gamma})$
$\displaystyle=\pi_{1}({\Gamma}_{L})\ast\pi_{1}({\Gamma}\setminus{\Gamma}_{L}).$

We also have the free factor decompositions

$\displaystyle f_{*}(\pi_{1}({\Gamma}_{R}))$
$\displaystyle=\pi_{1}({\Gamma}_{L})\ast B,\text{ and}$
$\displaystyle\pi_{1}({\Gamma}\setminus{\Gamma}_{L})$
$\displaystyle=f_{*}(\pi_{1}({\Gamma}\setminus{\Gamma}_{R}))\ast C,$

for some free factors $B$ and $C$ of $\pi_{1}({\Gamma})$. Putting together
these decompositions, we get:

$\displaystyle\pi_{1}({\Gamma})$ $\displaystyle=\pi_{1}({\Gamma}_{L})\ast
B\ast f_{*}(\pi_{1}({\Gamma}\setminus{\Gamma}_{R}))$
$\displaystyle\pi_{1}({\Gamma})$ $\displaystyle=\pi_{1}({\Gamma}_{L})\ast
f_{*}(\pi_{1}({\Gamma}\setminus{\Gamma}_{R}))\ast C.$

Therefore, we have $\operatorname{rk}(B)=\operatorname{rk}(C)$.

Translating these equalities, we compute:

$\displaystyle\Phi_{\mathcal{E}^{c}}(f)$
$\displaystyle=\operatorname{cork}(\pi_{1}({\Gamma}\setminus{\Gamma}_{L}),f_{*}(\pi_{1}({\Gamma}\setminus{\Gamma}_{R})))-\operatorname{cork}(\pi_{1}({\Gamma}\setminus{\Gamma}_{L}),\pi_{1}({\Gamma}\setminus{\Gamma}_{R}))$
$\displaystyle=\operatorname{cork}(f_{*}(\pi_{1}({\Gamma}_{R})),\pi_{1}({\Gamma}_{L}))-\operatorname{cork}(\pi_{1}({\Gamma}_{R}),\pi_{1}({\Gamma}_{L}))$
$\displaystyle=\operatorname{cork}(\pi_{1}({\Gamma}_{R}),g_{*}(\pi_{1}({\Gamma}_{L})))-\operatorname{cork}(\pi_{1}({\Gamma}_{R}),\pi_{1}({\Gamma}_{L}))$
$\displaystyle=\Phi_{\mathcal{E}}(g)=-\Phi_{\mathcal{E}}(f),$

where the last equation follows from that $g$ is a proper inverse of $f$ and
$\Phi_{\mathcal{E}}$ is a homomorphism.

  2. (ii)

Let $f\in\operatorname{PPHE}({\Gamma})$. Choose an $x_{0}$ that determines a
partition that is compatible with both $A^{c}$ and $B^{c}$ as in the beginning
of this section. Then there exist admissible pairs
$({\Gamma}_{R_{A^{c}}},{\Gamma}_{L_{A^{c}}})$ and
$({\Gamma}_{R_{B^{c}}},{\Gamma}_{L_{B^{c}}})$ of $f$ with respect to $A^{c}$
and $B^{c}$ respectively. By taking small enough ${\Gamma}_{L_{A^{c}}}$ and
${\Gamma}_{L_{B^{c}}}$, we can ensure that ${\Gamma}_{R_{A^{c}}}$ and
${\Gamma}_{R_{B^{c}}}$ have contractible intersection in ${\Gamma}$; See
Figure 2.

Figure 2. Illustration of the choices of subgraphs for the proof of
Proposition 3.9 (ii). Here the paths from $x_{0}$ to each subgraph are
omitted. We can choose pairs of graphs
$({\Gamma}_{R_{A}^{c}},{\Gamma}_{L_{A}^{c}})$ and
$({\Gamma}_{R_{B}^{c}},{\Gamma}_{L_{B}^{c}})$ such that the graphs from
different pairs have contractible intersections.

Then we observe that
$({\Gamma}_{R_{A^{c}}}\cup{\Gamma}_{R_{B^{c}}},{\Gamma}_{L_{A^{c}}}\cup{\Gamma}_{L_{B^{c}}})$
is an admissible pair for $f$ with respect to $A^{c}\cap B^{c}=(A\sqcup
B)^{c}$ (still with the basepoint $x_{0}$). We then have a free decomposition

$\pi_{1}({\Gamma}_{R_{A^{c}}}\cup{\Gamma}_{R_{B^{c}}},x_{0})\cong\pi_{1}({\Gamma}_{R_{A^{c}}},x_{0})\ast\pi_{1}({\Gamma}_{R_{B^{c}}},x_{0}),$

and the same for
$\pi_{1}({\Gamma}_{L_{A^{c}}}\cup{\Gamma}_{L_{B^{c}}},x_{0})$. Finally, we
compute

$\displaystyle\Phi_{(A\sqcup B)^{c}}$
$\displaystyle=\operatorname{cork}\left(A_{R_{A^{c}}}\ast
A_{R_{B^{c}}},f_{*}(A_{L_{A^{c}}}\ast
A_{L_{B^{c}}})\right)-\operatorname{cork}\left(A_{R_{A^{c}}}\ast
A_{R_{B^{c}}},A_{L_{A^{c}}}\ast A_{L_{B^{c}}}\right)$
$\displaystyle=\left(\operatorname{cork}(A_{R_{A^{c}}},f_{*}(A_{L_{A^{c}}}))+\operatorname{cork}(A_{R_{B^{c}}},f_{*}(A_{L_{B^{c}}}))\right)$
$\displaystyle\hskip
20.0pt-\left(\operatorname{cork}(A_{R_{A^{c}}},A_{L_{A^{c}}})+\operatorname{cork}(A_{R_{B^{c}}},A_{L_{B^{c}}})\right)$
$\displaystyle=\left(\operatorname{cork}(A_{R_{A^{c}}},f_{*}(A_{L_{A^{c}}}))-\operatorname{cork}(A_{R_{A^{c}}},A_{L_{A^{c}}})\right)$
$\displaystyle\hskip
20.0pt+\left(\operatorname{cork}(A_{R_{B^{c}}},f_{*}(A_{L_{B^{c}}}))-\operatorname{cork}(A_{R_{B^{c}}},A_{L_{B^{c}}})\right)$
$\displaystyle=\Phi_{A^{c}}+\Phi_{B^{c}}.$

Finally we apply Part (i) to see that

$\Phi_{A\sqcup B}=-\Phi_{(A\sqcup
B)^{c}}=-\Phi_{A^{c}}-\Phi_{B^{c}}=\Phi_{A}+\Phi_{B}.\qed$

######  Remark 3.10.

We remark that by Proposition 3.9 (i) and Proposition 3.9 (ii), we can even
formally define the flux map with respect to the empty set or the whole set
$E_{\ell}$:

$\Phi_{\emptyset}:=\Phi_{A}-\Phi_{A}\equiv
0,\qquad\Phi_{E_{\ell}}:=\Phi_{A}+\Phi_{A^{c}}\equiv 0.$

This allows us to define a flux map for any clopen $\mathcal{E}\subset E$ by
$\Phi_{\mathcal{E}}=\Phi_{\mathcal{E}\cap E_{\ell}}$.

###  3.3. Flux zero maps

In this section we will prove the following characterization of flux zero
maps.

######  Theorem 3.11.

Let ${\Gamma}$ be a locally finite, infinite graph with
$|E_{\ell}({\Gamma})|\geq 2$, and $f\in\operatorname{PMap}({\Gamma})$. Then
$f\in\overline{\operatorname{PMap_{\it c}}({\Gamma})}$ if and only if
$\Phi_{\mathcal{E}}(f)=0$ for every clopen subset $\mathcal{E}$ of
$E(\Gamma)$.

We have proved the forward direction already in a previous paper.

######  Proposition 3.12 ([DHK23, Proposition 7.13]).

If $f\in\overline{\operatorname{PMap}_{c}({\Gamma})}$, then
$\Phi_{\mathcal{E}}(f)=0$ for every clopen subset $\mathcal{E}$ of
$E({\Gamma})$.

We will first assume that ${\Gamma}$ is a core graph, i.e.,
$E_{\ell}({\Gamma})=E({\Gamma})$. For brevity, we will temporarily drop the
subscript $\ell$ for $E_{\ell}$ while we work under this assumption. To
leverage the algebraic information (flux 0) to obtain topological information
(homotopy equivalence), we need the following fact:

######  Lemma 3.13 ([Hat02, Proposition 1B.9]).

Let $X$ be a connected CW complex and let $Y$ be $K(G,1)$. Then every
homomorphism $\pi_{1}(X,x_{0})\to\pi_{1}(Y,y_{0})$ is induced by a continuous
map $(X,x_{0})\to(Y,y_{0})$ that is unique up to homotopy fixing $x_{0}$.

Recall that a graph is $K(F,1)$ for $F$ a free group (the fundamental group of
the graph). Now we prove a preliminary lemma to construct a compact
approximation of a proper homotopy equivalence.

######  Lemma 3.14.

Let $\mathcal{E}\subset E({\Gamma})$ be a nonempty proper clopen subset and
$f\in\operatorname{PMap}({\Gamma})$. If $\Phi_{\mathcal{E}}(f)=0$, then given
any compact $K\subset{\Gamma}$, there exists
$\psi\in\operatorname{PMap}({\Gamma})$ such that

  1. (i)

(Compact approximation) $\psi f^{-1}\in\mathcal{V}_{K}$,

  2. (ii)

(Truncation) there exist disjoint subgraphs ${\Gamma}_{\mathcal{E}}$, and
${\Gamma}_{\mathcal{E}^{c}}$ of ${\Gamma}$ with end spaces $\mathcal{E}$ and
$\mathcal{E}^{c}$ respectively, such that
$\psi|_{{\Gamma}_{\mathcal{E}}}=\operatorname{id}$ and
$\psi|_{{\Gamma}_{\mathcal{E}^{c}}}=f|_{{\Gamma}_{\mathcal{E}^{c}}}$, and

  3. (iii)

(Same flux) $\Phi_{\eta}(\psi)=\Phi_{\eta}(f)$ for every clopen subset
$\eta\subset E({\Gamma})\setminus\mathcal{E}$.

###### Proof.

Let $\\{{\Gamma}_{n}\\}_{n\in\mathbb{Z}}$ be as in the definition of
$\Phi_{\mathcal{E}}$, for some choice of basepoint $x_{0}$. Now, given
$f\in\operatorname{PMap}({\Gamma})$ and any $n$ there is some $m_{n}>n$ that
makes $(m_{n},n)$ into an admissible pair for $f$. See Figure 3.

Figure 3. Illustration of ${\Gamma}_{n}$ and ${\Gamma}_{m_{n}}$ for the flux
map $\Phi_{\mathcal{E}}$. Note $x_{0}$ is realizing a partition of ends
compatible with the partition $\mathcal{E}\sqcup\mathcal{E}^{c}$.

Since $\Phi_{\mathcal{E}}(f)=0$ we have

($\ast$)
$\displaystyle\operatorname{cork}(\pi_{1}({\Gamma}_{m_{n}},x_{0}),\pi_{1}({\Gamma}_{n},x_{0}))=\operatorname{cork}(\pi_{1}({\Gamma}_{m_{n}},f(x_{0})),f_{*}(\pi_{1}({\Gamma}_{n},x_{0})))$

for each $n\in\mathbb{Z}$. This allows us to define an isomorphism
$\Psi_{n}:\pi_{1}({\Gamma},x_{0})\rightarrow\pi_{1}({\Gamma},f(x_{0}))$ for
each $n$. Here we use the notation $G\setminus\setminus H$ to denote the
complementary free factor of $H$ in $G$. Define

$\displaystyle\Psi_{n}=\begin{cases}\operatorname{Id}&\text{on
$\pi_{1}(\Gamma,x_{0})\setminus\setminus\
\pi_{1}({\Gamma}_{m_{n}},x_{0})$},\\\ \sigma_{n}&\text{on
}\pi_{1}(\Gamma_{m_{n}},x_{0})\setminus\setminus\
\pi_{1}({\Gamma}_{n},x_{0}),\\\ f_{*}&\text{on
$\pi_{1}({\Gamma}_{n},x_{0})$},\end{cases}$

where $\sigma_{n}:\pi_{1}({\Gamma}_{m_{n}},x_{0})\setminus\setminus\
\pi_{1}({\Gamma}_{n},x_{0})\rightarrow\pi_{1}({\Gamma}_{m_{n}},f(x_{0}))\setminus\setminus\
f_{*}(\pi_{1}({\Gamma}_{n},x_{0}))$ is any isomorphism. Such $\sigma_{n}$ is
guaranteed to exist by ($\ast$ ‣ 3.3).

Now by 3.13, for each $n$ there exists a homotopy equivalence
$\psi_{n}:({\Gamma},x_{0})\to({\Gamma},f(x_{0}))$ such that

$\psi_{n}=\begin{cases}\operatorname{Id}&\text{on
${\Gamma}\setminus{\Gamma}_{m_{n}}$,}\\\ f&\text{on
${\Gamma}_{n}$}.\end{cases}$

Also note $\psi_{n}$ is a _proper_ homotopy equivalence, as it can be defined
in pieces as proper maps. Further, $\psi_{n}$ fixes the ends of ${\Gamma}$,
because $f$ does and ${\Gamma}_{m_{n}}\setminus{\Gamma}_{n}$ is compact. One
can similarly define its proper homotopy inverse. Hence, for each $n$ we have
$[\psi_{n}]\in\operatorname{PMap}({\Gamma})$.

The subgraphs $\\{{\Gamma}_{n}\\}_{n\in\mathbb{Z}}$ form an exhaustion of
${\Gamma}$, so $\psi_{n}\rightarrow f$ in $\operatorname{PMap}({\Gamma})$.
Therefore, for a compact $K\subset{\Gamma}$, there exists an
$n^{\prime}\in\mathbb{Z}$ such that
$\psi_{n^{\prime}}f^{-1}\in\mathcal{V}_{K}$. Take $\psi=\psi_{n^{\prime}}$ and
set ${\Gamma}_{\mathcal{E}^{c}}={\Gamma}_{n^{\prime}}$ and
${\Gamma}_{\mathcal{E}}=\overline{{\Gamma}\setminus{\Gamma}_{m_{n^{\prime}}}}$.
This gives (i) and (ii) by construction.

We now check that (iii) follows from (ii). Let $\eta$ be a clopen subset of
$E({\Gamma})$ that is disjoint from $\mathcal{E}$. We will actually check that
$\Phi_{\eta^{c}}(\psi)=\Phi_{\eta^{c}}(f)$. This will imply (iii) by
Proposition 3.9 (i).

Note $\eta\subset\mathcal{E}^{c}$. Now let ${\Gamma}_{m}$ be a subgraph from
the definition of $\Phi_{\eta^{c}}$ so that
${\Gamma}_{m}\subset{\Gamma}_{\mathcal{E}^{c}}$. Then there exists $n\leq m$
such that $(m,n)$ is admissible for $\psi$ with respect to the flux map
$\Phi_{\eta^{c}}$. Since $f=\psi$ on
${\Gamma}_{n}\subset{\Gamma}_{m}\subset{\Gamma}_{\mathcal{E}^{c}}$ by (ii), we
see that $\Phi_{\eta^{c}}(\psi)=\Phi_{\eta^{c}}(f)$ with the admissible pair
of graphs $({\Gamma}_{m},{\Gamma}_{n})$. ∎

######  Remark 3.15.

The reader may wonder why in the proof above we chose to define this sequence
of maps and argue via convergence in place of constructing the map $\psi$ by
hand as in [APV20]. While it is not too difficult to construct a $\psi$ so
that $\psi f^{-1}$ is the identity on a given compact $K$, it is significantly
more finicky to guarantee that $\psi f^{-1}$ preserves the complementary
components of $K$. The convergence argument given above allows us to avoid the
messy details of this.

######  Proposition 3.16.

Let ${\Gamma}$ be a locally finite, infinite graph with
$E({\Gamma})=E_{\ell}({\Gamma})$, $|E({\Gamma})|\geq 2$, and
$f\in\operatorname{PMap}({\Gamma})$. If $\Phi_{{\mathcal{E}}}(f)=0$ for every
clopen subset $\mathcal{E}$ of $E({\Gamma})$, then
$f\in\overline{\operatorname{PMap}_{c}({\Gamma})}$.

###### Proof.

Assume $f\in\operatorname{PMap}({\Gamma})$ has $\Phi_{\mathcal{E}}(f)=0$ for
every nonempty proper clopen subset $\mathcal{E}$ of the end space
$E({\Gamma})$. Given any compact $K\subset{\Gamma}$ we will find
$\psi\in\operatorname{PMap_{\it c}}({\Gamma})$ such that $\psi
f^{-1}\in\mathcal{V}_{K}$.

Without loss of generality we may enlarge $K$ so that it is connected, has at
least two complementary components, and every complementary component of $K$
is infinite. Then the complement of $K$ induces a partition of the ends. Write

$\mathcal{P}_{K}=\mathcal{E}_{1}\sqcup\ldots\sqcup\mathcal{E}_{n}$

for this partition.

Apply 3.14 to $f$ using $\mathcal{E}_{1}$ to obtain $\psi_{1}$. Note that by
(iii) we still have
$\Phi_{\mathcal{E}_{2}}(\psi_{1})=\Phi_{\mathcal{E}_{2}}(f)=0$. Thus we can
apply the lemma again to $\psi_{1}$ using $\mathcal{E}_{2}$ to obtain a
$\psi_{2}$. Continue this process recursively to obtain $\psi_{n}$.

Now, by (i) of 3.14, there exist $v_{1},\ldots,v_{n}\in\mathcal{V}_{K}$ such
that

$\displaystyle\psi_{i}$ $\displaystyle=\begin{cases}v_{i}\psi_{i-1}&\text{for
$1<i\leq n$,}\\\ v_{1}f&\text{for $i=1$}.\end{cases}$

Putting these together gives $\psi_{n}f^{-1}=v_{n}v_{n-1}\cdots
v_{1}\in\mathcal{V}_{K}$ as $\mathcal{V}_{K}$ is a subgroup.

It remains to check that $\psi_{n}\in\operatorname{PMap_{\it c}}({\Gamma})$.
However, by (ii), we have that $\psi_{n}$ is equal to the identity on
$\bigcup_{i=1}^{n}{\Gamma}_{\mathcal{E}_{i}}$. This exactly covers all of the
ends of ${\Gamma}$ as $\mathcal{P}_{K}$ was a partition of the ends. Therefore
we see that $\psi_{n}$ is supported on
$\overline{\bigcap_{i=1}^{n}{\Gamma}\setminus{\Gamma}_{\mathcal{E}_{i}}}$, a
compact set. Taking $\psi=\psi_{n}$ gives the desired compact approximation of
$f$.

Finally, since the $K$ above was taken to be arbitrary, starting with a
compact exhaustion of ${\Gamma}$ we can apply the above to obtain a sequence
of compactly supported maps that converge to $f$. ∎

Now we turn to the case where ${\Gamma}$ is not necessarily a core graph.

###### Proof of 3.11.

The forward direction follows from 3.12.

For the backward direction, we first homotope $f$ so that it fixes the
vertices of ${\Gamma}$. Then we see that we can write $f=f_{T}f_{c}$ where
$f_{T}$ has support on ${\Gamma}\setminus{\Gamma}_{c}$ and $f_{c}$ has support
on ${\Gamma}_{c}$.

We can see that $f_{T}\in\overline{\operatorname{PMap}_{c}({\Gamma})}$.
Indeed, enumerate the components of ${\Gamma}\setminus{\Gamma}_{c}$ as
$\\{R_{i}\\}_{i\in I}$ where each $R_{i}$ is a tree and $I$ is either finite
or $I=\mathbb{N}$. Then we can decompose $f_{T}=\prod_{i\in I}f_{i}$ where
each $f_{i}$ has compact support on $R_{i}$. Indeed, each has compact support
as $f_{T}$ is proper and thus the pre-image of the cutpoint
$\overline{R_{i}}\cap{\Gamma}_{c}$ is compact and $f_{i}$ can be homotoped to
have support contained within the convex hull of this full pre-image of the
cutpoint. Furthermore, all of the $f_{i}$ pairwise commute as each $f_{i}$ can
be homotoped so that it is totally supported away from the support of each
other $f_{j}$. Thus, we see that $f_{T}\in\overline{\operatorname{PMap_{\it
c}}({\Gamma})}$ as it is realized as the limit of partial products of the
$f_{i}$.

This also shows that given any flux map $\Phi_{\mathcal{E}}$ we must have that
$\Phi_{\mathcal{E}}(f_{T})=0$, again by 3.12. Therefore, given an
$\mathcal{E}$ with $\Phi_{\mathcal{E}}(f)=0$ we must have that
$\Phi_{\mathcal{E}}(f_{c})=0$ as $\Phi_{\mathcal{E}}$ is a homomorphism. We
can then apply 3.16 to conclude the desired result. ∎

###  3.4. Space of flux maps

Before we can prove B we need to endow the set of flux maps with an algebraic
structure. In the surface case, [APV20] could utilize the first integral
(co)homology of separating curves on the surface to give structure to the flux
maps they defined. Here we will be using the group of locally constant
$\mathbb{Z}$-valued functions on $E_{\ell}({\Gamma})$ in place of the homology
of separating curves. We remark that this is really the zeroth Čech cohomology
of $E_{\ell}({\Gamma})$ with coefficients in the constant sheaf $\mathbb{Z}$.
In Section 3.6 we observe that this perspective also works in the surface
case.

For a topological space $X$, we denote by $\check{C}(X)$ the group of locally
constant $\mathbb{Z}$-valued functions on $X$. The group operation is given by
addition of functions. We let $\mathring{C}(X)=\check{C}(X)/\mathbb{Z}$, the
quotient obtained by identifying the constant functions with zero. We will now
give a collection of some facts about $\mathring{C}(E)$ when $E$ is a compact,
totally disconnected, and metrizable space (i.e. a closed subset of a Cantor
set).

We identify the Cantor set,
$\mathcal{C}=2^{\mathbb{N}}=\\{0,1\\}^{\mathbb{N}}$, with the set of countable
binary sequences. A countable basis of clopen sets for the topology is then
given by the cylinder sets

$\displaystyle C_{a_{1}\cdots a_{k}}:=\\{(x_{n})\in 2^{\mathbb{N}}\ |\
x_{i}=a_{i},\;i=1,\ldots,k\\}$

where $a_{1}\cdots a_{k}$ is some finite binary sequence of length $k$. Say
such a cylinder set has width $k$. For $E$ a closed subset of the Cantor set
$\mathcal{C}$, a cylinder set of $E$ is the intersection of a cylinder set for
$\mathcal{C}$ with $E$, i.e., a set of the form $C_{w}\cap E$ where $w\in
2^{k}$ for some $k\geq 0$. The standard tree model for the Cantor set is the
usual rooted binary tree, and for an arbitrary closed subset
$E\subset\mathcal{C}$ we take the subtree with the end space $E$. Given a
subset, $A$, of a topological space we let $\chi_{A}$ denote the indicator
function on $A$.

######  Theorem 3.17 (Countable Basis for $\mathring{C}(E)$).

Let $E$ be a compact, totally disconnected, and metrizable space. There exists
a countable collection $\mathcal{A}=\\{A_{i}\\}_{i\in I}$ of cylinder sets of
$E$ so that

  1. (1)

Any cylinder set $C$ of $E$ that is not in $\mathcal{A}$ can be written as
$C=A_{0}\setminus(A_{1}\sqcup\cdots\sqcup A_{n})$ for some
$A_{0}\in\mathcal{A}$, and some $A_{j}\in\mathcal{A}$, with $A_{j}\subset
A_{0}$ and $A_{j}\cap A_{k}=\emptyset$ for all distinct
$j,k\in\\{1,\ldots,n\\}$,

  2. (2)

$\mathcal{B}=\\{\chi_{A_{i}}\\}_{i\in I}$ is a free basis for
$\mathring{C}(E)$. In particular, $\mathring{C}(E)=\oplus_{i\in I}\mathbb{Z}$,
and

  3. (3)

for $T$ the standard tree model of the end space $(E,\emptyset)$, there exists
an injective map $\iota:\mathcal{A}\rightarrow T$ so that $\iota$ maps into
the interior of edges and $\iota(\mathcal{A})$ cuts the graph into a
collection of one ended graphs.

###### Proof.

Note that if $|E|=n<\infty$ then the result is immediate by taking
$\mathcal{A}$ to be the collection of all individual ends except one. Hence,
we will assume that $E$ is infinite.

We first prove the result for $E=\mathcal{C}$ the Cantor set. We define
$\mathcal{A}^{\prime}$ to be the set of all cylinder sets consisting of
cylinders of the form $C_{a_{1}\cdots a_{k-1}0}$ together with the whole space
$\mathcal{C}$. That is,

$\displaystyle\mathcal{A}^{\prime}=\\{\mathcal{C},C_{0},C_{00},C_{10},C_{000},C_{100},C_{010},C_{110},\ldots\\}$

We claim that $\\{\chi_{A}\\}_{A\in\mathcal{A}^{\prime}}$ forms a free basis
for $\check{C}(\mathcal{C})$. We first have

######  Claim.

For every $f\in\check{C}(\mathcal{C})$, there exist finitely many disjoint
clopen subsets $B_{1},\ldots,B_{n}$ and integers $b_{1},\ldots,b_{n}$ such
that

$f=\sum_{j=1}^{n}b_{j}\chi_{B_{j}}.$

###### Proof.

Suppose $f$ is a locally constant function on $\mathcal{C}$ with _infinitely
many_ distinct $\mathbb{Z}$-values $b_{1},b_{2},\ldots$. Then
$\\{f^{-1}(b_{j})\\}_{j=1}^{\infty}$ forms a clopen cover of $\mathcal{C}$
which does not have a finite subcover, contradicting the compactness of
$\mathcal{C}$. Therefore, $f$ can assume at most finitely different values in
$\mathbb{Z}$, and taking $B_{j}=f^{-1}(b_{j})$ proves the claim. ∎

Thus we can check that $\\{\chi_{A}\\}_{A\in\mathcal{A}^{\prime}}$ generates
$\check{C}(\mathcal{C})$ by verifying that for an arbitrary clopen set $B$ of
$\mathcal{C}$, we can write $\chi_{B}$ as a finite linear combination of
elements from $\\{\chi_{A}\\}_{A\in\mathcal{A}^{\prime}}$. Since the cylinder
sets form a clopen basis for the topology, we only need to check when $B$ is a
cylinder set. Take $B=C_{a_{1}\cdots a_{k}}$ for some $k>0$ and $a_{1}\cdots
a_{k}\in 2^{k}$. Then we have either $B\in\mathcal{A}^{\prime}$ or $a_{k}=1$.
Supposing the latter, let

$m=\begin{cases}0&\text{if $a_{1}=\ldots=a_{k}=1$,}\\\
\max\\{j|a_{j}=0\\}&\text{otherwise}\end{cases}$

Then we can write

$\displaystyle\chi_{B}=\chi_{C_{a_{1}\cdots a_{k}}}=\chi_{C_{a_{1}\cdots
a_{m}}}-\left(\sum_{j=m}^{k-1}\chi_{C_{a_{1}\cdots a_{j}0}}\right),$

where we take $a_{1}\cdots a_{m}$ as an empty sequence when $m=0$. Thus we see
that $\\{\chi_{A}\\}_{A\in\mathcal{A}^{\prime}}$ generates
$\check{C}(\mathcal{C})$. This also shows that property (1) holds.

Next we verify that the set
$\mathcal{B}^{\prime}:=\\{\chi_{A}\\}_{A\in\mathcal{A}^{\prime}}$ is linearly
independent. Suppose

$\displaystyle 0=\sum_{j=1}^{n}a_{j}\chi_{A_{j}},$

for some distinct $A_{1},\ldots,A_{n}\in\mathcal{A}^{\prime}$. We will proceed
by induction on $n$. The case when $n=1$ is straightforward. Now let $n>1$ and
without loss of generality we can assume that $A_{n}$ is of minimal width. Let
$w$ be the word defining $A_{n}$, i.e. $A_{n}=C_{w}$. Note that $w$ may be the
empty word (when $A_{n}=\mathcal{C})$. Consider the sequence $w\bar{1}$
consisting of the starting word $w$ followed by the constant infinite sequence
of $1$s. Then by minimality of $w$, we have

$\displaystyle 0=\sum_{j=1}^{n}a_{j}\chi_{A_{j}}(w\bar{1})=a_{n}.$

Therefore, we have
$0=\sum_{j=1}^{n}a_{j}\chi_{A_{j}}=\sum_{j=1}^{n-1}a_{j}\chi_{A_{j}}$ so by
the induction on $n$ we see that $a_{j}=0$ for all $j$. Thus we see that
$\mathcal{B}^{\prime}$ is a free basis for $\check{C}(\mathcal{C})$. Taking
$\mathcal{A}:=\mathcal{A}^{\prime}\setminus\\{\mathcal{C}\\}=\\{C_{0},C_{00},C_{10},C_{000},C_{100},C_{010},C_{110},\ldots\\}$,
the free basis $\mathcal{B}^{\prime}$ for $\check{C}(\mathcal{C})$ descends
(allowing for a slight abuse of notation) to a free basis
$\mathcal{B}:=\\{\chi_{A}\\}_{A\in\mathcal{A}}$ for
$\mathring{C}(\mathcal{C})$, proving (2).

Finally, we can define $\iota:\mathcal{A}\rightarrow T$ by using the labels on
each of the cylinder sets to map each cylinder set to the midpoint of its
corresponding edge in the standard binary tree model of the Cantor set. See
Figure 4 for a picture of the map. The components of
$T\setminus\iota(\mathcal{A})$ each contains one end from $T$.

$0$
---
$00$
---
$000$
---
$10$
---
$100$
---
$010$
---
$110$
---
Figure 4. The image of the map $\iota:\mathcal{A}\to T$ is given in blue.

Now to go from the Cantor set to a general infinite end space we identify $E$
with a subspace of $\mathcal{C}$ and take $\mathcal{A}=\\{C_{0}\cap
E,C_{00}\cap E,C_{10}\cap E,\ldots\\}$, deleting duplicated sets if necessary.
Then the set $\\{\chi_{A}\\}_{A\in\mathcal{A}}$ will still determine a free
basis for $\mathring{C}(E)$. ∎

Apply this theorem to $E_{\ell}({\Gamma})$ in order to obtain the set
$\mathcal{A}=\\{A_{i}\\}_{i\in I}$. We now define the homomorphism

$\displaystyle\Pi:\operatorname{PMap}({\Gamma})$
$\displaystyle\rightarrow\prod_{i\in I}\mathbb{Z}$ $\displaystyle f$
$\displaystyle\mapsto(\Phi_{A_{i}}(f))_{i\in I}.$

We will check that this map is surjective and has kernel exactly
$\overline{\operatorname{PMap_{\it c}}({\Gamma})}$, i.e. it forms the
following short exact sequence:

$\displaystyle 1\longrightarrow\overline{\operatorname{PMap_{\it
c}}({\Gamma})}\longrightarrow\operatorname{PMap}({\Gamma})\overset{\Pi}{\longrightarrow}\prod_{i\in
I}\mathbb{Z}\longrightarrow 1.$

######  Lemma 3.18.

Let $\mathcal{E}$ be a clopen subset of $E({\Gamma})$ so that $\mathcal{E}\cap
E_{\ell}({\Gamma})$ is a proper nontrivial subset. If
$f\in\operatorname{PMap}({\Gamma})$ satisfies $\Phi_{A}(f)=0$ for all
$A\in\mathcal{A}$, then $\Phi_{\mathcal{E}}(f)=0$ as well.

###### Proof.

We first note that $\mathcal{E}$ can be written as a disjoint union of
finitely many cylinder sets. Thus, by Proposition 3.9 (ii) it suffices to
check when $\mathcal{E}$ is a cylinder set $C$ of $E({\Gamma})$. Assume that
$f\in\operatorname{PMap}({\Gamma})$ satisfies $\Phi_{A_{i}}(f)=0$ for all
$i\in I$. Then $C\cap E_{\ell}({\Gamma})$ is again a cylinder set of
$E_{\ell}$. Applying property (1) of 3.17 we have either $C\in\mathcal{A}$, or
$C=A_{0}\setminus(\bigsqcup_{j=1}^{n}A_{j})$ for some $A_{0}\in\mathcal{A}$
and $A_{j}\in\mathcal{A}$. If $C\in\mathcal{A}$, then we conclude
$\Phi_{C}(f)=0$. For the other case, we can apply Proposition 3.9 (ii) and
Proposition 3.9 (iii) to write

$\displaystyle\Phi_{C}(f)=\Phi_{A_{0}}(f)-\sum_{j=1}^{n}\Phi_{A_{j}}(f)=0-0=0.$

∎

######  Corollary 3.19.

For ${\Gamma}$ and $\Pi$ as above,
$\ker(\Pi)=\overline{\operatorname{PMap_{\it c}}({\Gamma})}$.

###### Proof.

The forward direction of 3.11 implies
$\ker(\Pi)\supset\overline{\operatorname{PMap_{\it c}}({\Gamma})}$. On the
other hand, 3.18 together with the backward direction of 3.11 imply the other
containment $\ker(\Pi)\subset\overline{\operatorname{PMap_{\it
c}}({\Gamma})}$. ∎

Next, we will build a section to show $\Pi$ is surjective, and more
importantly, this sequence splits. This gives us our desired semidirect
product decomposition in B.

######  Proposition 3.20.

There exists an injective homomorphism $\hat{\iota}:\prod_{i\in
I}\mathbb{Z}\rightarrow\operatorname{PMap}({\Gamma})$ so that
$\Pi\circ\hat{\iota}$ is the identity on $\prod_{i\in I}\mathbb{Z}$.

###### Proof.

Let $T$ be the maximal tree of the graph ${\Gamma}_{c}$ in standard form. Note
that the end space of $T$ is homeomorphic to $E_{\ell}({\Gamma})$ and let
$\mathcal{A}=\\{A_{i}\\}_{i\in I}$ be the set obtained from (2) of 3.17
applied to the set $E_{\ell}({\Gamma})$ and $\iota:\mathcal{A}\rightarrow T$
be the map given by property (3) of 3.17. The closure in ${\Gamma}_{c}$ of
every complementary component of $\iota(\mathcal{A})$ is a one-ended subgraph
with infinite rank. Call one such component ${\Gamma}^{\prime}$. It has at
most a countably infinite number of half edges coming from the points of
$\iota(\mathcal{A})$. Now we will modify ${\Gamma}^{\prime}$ via a proper
homotopy equivalence that fixes $\partial{\Gamma}^{\prime}$ so that the new
graph has a “grid of loops” above $\partial{\Gamma}^{\prime}$. See Figure 5
for how this replacement is done. Such a replacement by a proper homotopy
equivalence is possible by the classification of infinite graphs.

$\iota(A_{i_{1}})$
---
$\iota(A_{i_{2}})$
---
$\iota(A_{i_{3}})$
---
$\iota(A_{i_{4}})$
---
$\iota(A_{i_{1}})$
---
$\iota(A_{i_{2}})$
---
$\iota(A_{i_{3}})$
---
$\iota(A_{i_{4}})$
---
$\iota(A_{i_{1}})$
---
$\iota(A_{i_{2}})$
---
$\iota(A_{i_{3}})$
---
$\iota(A_{i_{1}})$
---
$\iota(A_{i_{2}})$
---
$\iota(A_{i_{3}})$
---
Figure 5. The new replacement graphs for each component of
$T\setminus\iota(\mathcal{A})$. The top picture shows the case when a
component has infinitely many cut points and the bottom for finitely many.
Note that above each cut point one sees a “column of loops” within the grid.

After replacing each component of ${\Gamma}_{c}\setminus\iota(\mathcal{A})$ we
obtain a new graph that is proper homotopy equivalent to the original
${\Gamma}_{c}$. We can also extend this proper homotopy equivalence to the
entire graph ${\Gamma}$, as our proper homotopy equivalence fixes the boundary
points of each complementary component of $\iota(\mathcal{A})$. Now for each
$i$, there are exactly two complementary components whose closures in
${\Gamma}_{c}$ contain $\iota(A_{i})$. Let
$\ell_{i}\in\operatorname{PMap}({\Gamma})$ be the loop shift supported on the
two columns of loops sitting above $\iota(A_{i})$ in these components. Orient
the loop shift so that it is shifting towards the end in $A_{i}$.

Note that each $\ell_{i}$ has total support disjoint from each other
$\ell_{j}$ so that $\ell_{i}\ell_{j}=\ell_{j}\ell_{i}$ for all $i,j\in I$.
Therefore, $\prod_{i\in
I}\langle\ell_{i}\rangle<\operatorname{PMap}({\Gamma})$, and we can define the
homomorphism $\hat{\iota}:\prod_{i\in
I}\mathbb{Z}\to\operatorname{PMap}({\Gamma})$ by

$\displaystyle\hat{\iota}\left((n_{i})_{i\in I}\right):=\prod_{i\in
I}\ell_{i}^{n_{i}}.$

It remains to check that $\Pi\circ\hat{\iota}$ is the identity on $\prod_{i\in
I}\mathbb{Z}$. By the construction of the loop shifts, $\ell_{i}$ crosses
exactly one of the clopen subsets in $\mathcal{A}$, namely $A_{i}$. Therefore,
we have

$\displaystyle\Phi_{A_{j}}(\ell_{i})=\delta_{ij}:=\begin{cases}1\;\;\text{ if
}i=j,\\\ 0\;\;\text{ if }i\neq j.\end{cases}$

Now, given any tuple $(n_{i})_{i\in I}\in\prod_{i\in I}\mathbb{Z}$ we compute

$\displaystyle(\Pi\circ\hat{\iota})\left((n_{i})_{i\in I}\right)$
$\displaystyle=\Pi\left(\prod_{i\in
I}\ell_{i}^{n_{i}}\right)=\left(\Phi_{A_{j}}\left(\prod_{i\in
I}\ell_{i}^{n_{i}}\right)\right)_{j\in I}=(n_{i})_{i\in I}.\qed$

###### Proof of B.

3.19 and 3.20 above give the desired splitting short exact sequence
$1\longrightarrow\overline{\operatorname{PMap}_{c}({\Gamma})}\longrightarrow\operatorname{PMap}({\Gamma})\longrightarrow\mathbb{Z}^{\alpha}\longrightarrow
1$, with $\alpha=|\mathcal{A}|-1$. ∎

###  3.5. The rank of integral cohomology

As pointed out in 3.10, we define $\Phi_{\emptyset}=\Phi_{E_{\ell}}\equiv 0$.

######  Lemma 3.21.

Let $\\{A_{i}\\}_{i\in I}$ be a clopen collection of subsets of
$E_{\ell}({\Gamma})$ such that $\mathcal{B}=\\{\chi_{A_{i}}\\}_{i\in I}$ is a
free basis for $\mathring{C}(E_{\ell})$ as in 3.17. Then the map

$\displaystyle\Theta:\mathring{C}(E_{\ell}({\Gamma}))$
$\displaystyle\longrightarrow
H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z}),$ $\displaystyle\sum_{i\in
I}{n_{i}}\chi_{A_{i}}$ $\displaystyle\longmapsto\sum_{i\in
I}{n_{i}}\Phi_{A_{i}}.$

is a well-defined injective homomorphism.

###### Proof.

Since $\mathcal{B}$ is a free basis for $\mathring{C}(E_{\ell})$, the map
$\chi_{A_{i}}\mapsto\Phi_{A_{i}}$ on $\mathcal{B}$ extends to a well-defined
homomorphism on the whole group $\mathring{C}(E_{\ell})$. To see $\Theta$ is
injective, suppose
$\Theta(\sum_{i}n_{i}\chi_{A_{i}})=\sum_{i}n_{i}\Phi_{A_{i}}=0$ for
$\chi_{A_{i}}\in\mathcal{B}$. Then for each $j$ that arises as an index of the
summation, we evaluate the sum at the loop shift $\ell_{j}$ constructed in the
proof of 3.20:

$0=\sum_{i}n_{i}\Phi_{A_{i}}(\ell_{j})=n_{j}\Phi_{A_{j}}(\ell_{j})=n_{j},$

which implies that $\sum_{i}n_{i}\chi_{A_{i}}\equiv 0$, concluding that
$\Theta$ is injective. ∎

Here we collect relevant results on the first homology of the pure mapping
class group of graphs of rank $n$ with $s$ rays.

######  Fact 3.22 ([HV98, Theorem 1.1]).

$H_{1}(\operatorname{Aut}(F_{n});\mathbb{Q})=0$ for all $n\geq 1$.

######  Fact 3.23 ([Hat95, Section 4]).

For $n\geq 3$ and $s\geq 1$,

$H_{1}(F_{n}^{s-1}\rtimes\operatorname{Aut}(F_{n});\mathbb{Z})\cong
H_{1}(F_{n}^{s}\rtimes\operatorname{Aut}(F_{n});\mathbb{Z}).$

This still holds for $n=1,2$ if $s\geq 2$.

######  Proposition 3.24.

$H^{1}(\operatorname{PMap}_{c}({\Gamma});\mathbb{Z})=0$ for every locally
finite, infinite graph ${\Gamma}$.

###### Proof.

Let $\\{{\Gamma}_{k}\\}$ be a compact exhaustion of ${\Gamma}$. Then
$\operatorname{PMap}_{c}({\Gamma})$ is a direct limit of
$\operatorname{PMap}({\Gamma}_{k})$’s, each of which is isomorphic to
$F_{n_{k}}^{e_{k}}\rtimes\operatorname{Aut}(F_{n_{k}})$ for some $e_{k}\geq 0$
and $n_{k}\geq 1$ (Recall 2.4). Since the direct limit commutes with
$H^{1}(-;\mathbb{Z})\equiv\operatorname{Hom}(-,\mathbb{Z})$, it suffices to
show that groups of the form $F_{n}^{e}\rtimes\operatorname{Aut}(F_{n})$ have
trivial first cohomology. We first show
$H^{1}(\operatorname{Aut}(F_{n});\mathbb{Z})=0$. By the universal coefficient
theorem for cohomology,

$0\longrightarrow\operatorname{Ext}\left(H_{0}(\operatorname{Aut}(F_{n});\mathbb{Q}),\mathbb{Z}\right)\longrightarrow
H^{1}(\operatorname{Aut}(F_{n});\mathbb{Z})\longrightarrow\operatorname{Hom}(H_{1}(\operatorname{Aut}(F_{n});\mathbb{Q}),\mathbb{Z})\longrightarrow
0$

where
$\operatorname{Ext}\left(H_{0}(\operatorname{Aut}(F_{n});\mathbb{Q});\mathbb{Z}\right)=0$
as $H_{0}(\operatorname{Aut}(F_{n});\mathbb{Q})\cong\mathbb{Q}$ is free. Also,
$H_{1}(\operatorname{Aut}(F_{n});\mathbb{Q})=0$ by 3.22, so it follows that
$H^{1}(\operatorname{Aut}(F_{n});\mathbb{Z})=0$.

On the other hand, repeatedly applying 3.23 together with the universal
coefficient theorem for homology shows that for $n\geq 3$,

$H_{1}(F_{n}^{s}\rtimes\operatorname{Aut}(F_{n});\mathbb{Q})=H_{1}(F_{n}^{s-1}\rtimes\operatorname{Aut}(F_{n});\mathbb{Q})=\ldots=H_{1}(\operatorname{Aut}(F_{n});\mathbb{Q})=0.$

The last equality comes from 3.22. For $n=1,2$, the argument is the same,
except we reduce the problem of showing
$H^{1}(F_{n}^{s-1}\rtimes\operatorname{Aut}(F_{n});\mathbb{Z})=0$ to checking
$H_{1}(F_{n}\rtimes\operatorname{Aut}(F_{n});\mathbb{Q})=0$. One can check
$\mathbb{Z}\rtimes\mathbb{Z}_{2}$ and $F_{2}\rtimes\operatorname{Aut}(F_{2})$
have finite abelianization to conclude this. (See e.g. [AFV08, Corollary 2]
for a finite presentation of $\operatorname{Aut}(F_{2})$.) This completes the
proof of $H^{1}(\operatorname{PMap}_{c}({\Gamma});\mathbb{Z})=0$. ∎

######  Theorem 3.25.

The map $\Theta$ in 3.21 is an isomorphism.

###### Proof.

We only need to check the surjectivity of $\Theta$. Pick $\phi\in
H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})=\operatorname{Hom}(\operatorname{PMap}({\Gamma}),\mathbb{Z})$.
By 3.24, we have $\phi(\operatorname{PMap_{\it c}}({\Gamma}))=\\{0\\}$. By
Dudley’s automatic continuity [Dud61], $\phi$ is continuous, so
$\phi(\overline{\operatorname{PMap_{\it c}}({\Gamma})})=\\{0\\}$. Recall the
semidirect product decomposition
$\operatorname{PMap}({\Gamma})\cong\overline{\operatorname{PMap}_{c}({\Gamma})}\rtimes
L$ from B, where $L\cong\prod_{i\in I}\langle\ell_{i}\rangle$, the product of
commuting loop shifts. Furthermore, these loop shifts are dual to the
collection of $\\{\Phi_{A_{i}}\\}_{i\in I}\subset
H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})$ so that
$\Phi_{A_{j}}(\ell_{i})=\delta_{ij}$. Since $\phi$ is zero on the
$\overline{\operatorname{PMap}_{c}({\Gamma})}$-factor, it follows that $\phi$
is completely determined by its value on $L$. Note also that
$L\cong\prod_{i\in I}\mathbb{Z}$ so that $H^{1}(L;\mathbb{Z})\cong\oplus_{i\in
I}\mathbb{Z}$ where a basis for $H^{1}(L;\mathbb{Z})$ is given exactly by the
set $\\{\Phi_{A_{i}}\\}_{i\in I}$, as in 3.17(2). Hence, $\phi=\phi|_{L}\in
H^{1}(L;\mathbb{Z})$ can be described by a finite linear combination of
$\Phi_{A_{i}}$’s. Such a finite linear combination is the image of a finite
linear combination of $\chi_{A_{i}}$ under $\Theta$, so $\Theta$ is
surjective. ∎

######  Corollary 3.26 (C, revisited).

For every locally finite, infinite graph ${\Gamma}$,

$\operatorname{rk}\left(H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})\right)=\begin{cases}0&\text{if
$|E_{\ell}|\leq 1$}\\\ n-1&\text{if $2\leq|E_{\ell}|=n<\infty$}\\\
\aleph_{0}&\text{otherwise}.\end{cases}$

###### Proof.

This follows from the isomorphism
$\Theta:\mathring{C}(E_{\ell}({\Gamma}))\cong
H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})$ in 3.25. ∎

###  3.6. Relation to surfaces

Aramayona–Patel–Vlamis in [APV20] obtain a result similar to 3.25 in the
infinite-type _surface_ case using the homology of separating curves in place
of $\mathring{C}(E_{\ell}({\Gamma}))$. Here we show that these approaches can
be unified, as they each rely solely on the subspace of ends accumulated by
loops or genus. Let $S$ be an infinite-type surface and let $\hat{S}$ be the
surface obtained from $S$ by forgetting the planar ends of $S$. Let
$H_{1}^{sep}(\hat{S};\mathbb{Z})$ be the subgroup of
$H_{1}(\hat{S};\mathbb{Z})$ generated by homology classes that have separating
simple closed curves of $\hat{S}$ as representatives. Note that when $S$ has
only planar ends, $H_{1}^{sep}(\hat{S};\mathbb{Z})$ is trivial.

######  Theorem 3.27 ([APV20, Theorem 4] for genus $\geq 2$, [DP20, Theorem
1.1] for genus $1$).

Let $S$ be an infinite-type surface of genus at least one. Then

$\displaystyle H^{1}(\operatorname{PMap}(S);\mathbb{Z})\cong
H_{1}^{sep}(\hat{S};\mathbb{Z}).$

Let $E_{g}(S)$ denote the space of ends of $S$ accumulated by genus (i.e., the
non-planar ends).

######  Proposition 3.28.

Let $S$ be an infinite-type surface. Then

$\displaystyle H_{1}^{sep}(\hat{S};\mathbb{Z})\cong\mathring{C}(E_{g}(S)).$

###### Proof.

We first note that by definition, $E_{g}(S)=E(\hat{S})$. Let $v\in
H_{1}^{sep}(\hat{S};\mathbb{Z})$ be a primitive element, i.e. $v$ has a
representative $\gamma$ that is an oriented and separating simple closed
curve. Now $v$ determines a partition of $E(\hat{S})$ into two clopen subsets,
$v^{+}$, those ends to the right of $\gamma$, and $v^{-}$, those ends to the
left of $\gamma$. Note that these are proper subsets if and only if $v\neq 0$
if and only if $\chi_{v^{+}}\neq 0$ in $\mathring{C}(E)$. Define

$\displaystyle\Xi(v):=\chi_{v^{+}}\in\mathring{C}(E),$

for each nonzero primitive element $v$ of $H_{1}^{sep}(\hat{S};\mathbb{Z})$.
This linearly extends to define an isomorphism
$\Xi:H_{1}^{sep}(\hat{S};\mathbb{Z})\xrightarrow{\sim}\mathring{C}(E_{g}(S))$.
∎

######  Corollary 3.29.

Let $S$ be an infinite-type surface of genus at least one and ${\Gamma}$ be a
locally finite, infinite graph. If $E_{g}(S)$ is homeomorphic to
$E_{\ell}({\Gamma})$, then there is a natural isomorphism between
$H^{1}(\operatorname{PMap}(S);\mathbb{Z})$ and
$H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})$.

###### Proof.

We first note that if $E_{g}(S)$ is empty (i.e. $S$ has finite genus), then
$H^{1}(\operatorname{PMap}(S);\mathbb{Z})$ is trivial by [APV20, Theorem 1]
and [DP20, Theorem 1.1]. Similarly, if $E_{\ell}(S)$ is empty, then
$H^{1}(\operatorname{PMap}(S);\mathbb{Z})$ is trivial by 3.2 and 3.24.

Otherwise, the isomorphism is obtained by composing the maps from 3.25, 3.27,
and 3.28:

$H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})\overset{\Theta}{\cong}\mathring{C}(E_{\ell}({\Gamma}))\cong\mathring{C}(E_{g}(S))\overset{\Xi}{\cong}H_{1}^{sep}(\hat{S};\mathbb{Z})\cong
H^{1}(\operatorname{PMap}(S);\mathbb{Z}).$

∎

##  4\. CB generation classification

As an application of 3.11 and B, in this section we obtain A, the
classification of infinite graphs with CB generating sets. We revisit the
theorem for convenience.

######  Theorem 4.1 (A, revisited).

Let ${\Gamma}$ be a locally finite, infinite graph. Then
$\operatorname{PMap}({\Gamma})$ is CB generated if and only if either
${\Gamma}$ is a tree, or satisfies the following:

  1. (1)

${\Gamma}$ has finitely many ends accumulated by loops, and

  2. (2)

there is no accumulation point in $E\setminus E_{\ell}$.

The only if direction of A comes from [DHK23]:

######  Proposition 4.2 ([DHK23, Theorem 6.1]).

Let ${\Gamma}$ be a locally finite, infinite graph. If
$\operatorname{rk}({\Gamma})>0$ and $E\setminus E_{\ell}$ has an accumulation
point, then $\operatorname{PMap}({\Gamma})$ is not CB-generated.

######  Proposition 4.3 ([DHK23, Theorem 8.2]).

Let ${\Gamma}$ be a locally finite, infinite graph. If ${\Gamma}$ has
infinitely many ends accumulated by loops, then
$\operatorname{PMap}({\Gamma})$ is not locally CB. In particular,
$\operatorname{PMap}({\Gamma})$ is not CB-generated.

Now we show that those conditions are also sufficient for CB-generation.
First, recall by 2.6 that when ${\Gamma}$ is a tree,
$\operatorname{PMap}({\Gamma})$ is the trivial group. We proceed to show (1)
and (2) are sufficient to show that $\operatorname{PMap}({\Gamma})$ is CB
generated. We start with the case where ${\Gamma}$ has finite rank and
satisfies Condition (2):

######  Proposition 4.4.

Let ${\Gamma}$ be a locally finite, infinite graph. If ${\Gamma}$ has finite
rank with no accumulation point in $E$, then $\operatorname{PMap}({\Gamma})$
is finitely generated.

###### Proof.

Note in this case $E_{\ell}$ is the empty set, so having no accumulation point
in $E\setminus E_{\ell}$ is equivalent to having a finite end space. Hence
$\operatorname{PMap}({\Gamma})$ is isomorphic to one of
$\operatorname{Out}(F_{n}),\operatorname{Aut}(F_{n})$ or
$F_{n}^{e}\rtimes\operatorname{Aut}(F_{n})$ for some $e=|E|-1\geq 1$, all of
which are finitely generated, concluding the proof. ∎

Now assume ${\Gamma}$ has infinite rank but finitely many ends accumulated by
loops with no accumulation point in $E\setminus E_{\ell}$. As in 1.1,
${\Gamma}$ can be realized as a finite wedge sum of rays, Loch Ness monster
graphs (infinite rank graph with end space
$(E,E_{\ell})\cong(\\{*\\},\\{*\\})$), and Millipede monster graphs (infinite
rank graph with end space
$(E,E_{\ell})\cong(\mathbb{N}\cup\\{\infty\\},\\{\infty\\})$). Then ${\Gamma}$
is characterized by the triple $(r,l,m)$ where $r$ is the number of ray
summands, $l$ is the number of Loch Ness monster summands, and $m$ is the
number of Millipede monster summands. Then ${\Gamma}$ is as in Figure 6. Note
that this triple is not unique, in fact, if $m>0$ then we do not need to keep
track of $r$ as any additional ray can simply be moved via a proper homotopy
into a Millipede monster summand. However, in order to avoid a case-by-case
analysis we prove that $\operatorname{PMap}({\Gamma})$ is CB-generated for
_any_ triple $(r,l,m)$. Note that we already know by 2.16 that both the Loch
Ness monster graph, $(0,1,0)$, and the Millipede monster graph, $(0,0,1)$,
have CB and thus CB-generated pure mapping class groups. Therefore we will
ignore these two graphs throughout this section.

$l$
---
$m$
---
$r$
---
$K$
---
Figure 6. The graphs ${\Gamma}$ that we prove have a CB-generating pure
mapping class group. Each such ${\Gamma}$ has a single wedge point $K$ and
${\Gamma}\setminus K$ has $r$ ray components, $l$ Loch Ness monster
components, and $m$ Millipede monster components.

The foundation for our choice of CB-generating set for
$\operatorname{PMap}({\Gamma})$ will be the set $\mathcal{V}_{K}$, where $K$
is the wedge point as in Figure 6. Recall that an appropriate choice of a
compact set $K$ provides a CB neighborhood of the identity certifying that
$\operatorname{PMap}({\Gamma})$ is locally CB.

######  Proposition 4.5 ([DHK23, Proposition 8.3]).

Let ${\Gamma}$ be a locally finite, infinite graph with finitely many ends
accumulated by loops. Then $\operatorname{PMap}({\Gamma})$ is locally CB if
and only if ${\Gamma}\setminus{\Gamma}_{c}$ has only finitely many components
whose ends space is infinite. Moreover, for any choice of connected compact
subgraph $K$ whose complementary components are either trees or monster
graphs, $\mathcal{V}_{K}$ is a CB neighborhood of the identity in
$\operatorname{PMap}({\Gamma})$.

We remark that the moreover statement is absent in [DHK23, Proposition 8.3];
however, it can be deduced readily from the proof. We thus have that our
choice of $\mathcal{V}_{K}$ is CB. This is the starting point for our CB
generating set; we now describe how to choose the remaining elements.

Enumerate each of the ray summands of ${\Gamma}$ as $R_{1},\ldots,R_{r}$, the
Loch Ness monster summands as $L_{1},\ldots,L_{l}$, and the Millipede monster
summands as $M_{1},\ldots,M_{m}$ (skip the enumeration if there are no
summands of a given type). We also sequentially label the loops in $L_{i}$ by
$a_{i,j}$ where $a_{i,1}$ is the loop closest to $K$. We similarly label the
loops in $M_{i}$ by $b_{i,j}$. For each $R_{i}$ let $I_{i}$ be an interval in
the interior of $R_{i}$. Then we have the following finite collection of word
maps:

$\displaystyle W:=\\{\phi_{(a_{1,1},I_{i})}\\}_{i=1}^{r}.$

If $l=0$ then we use $W:=\\{\phi_{(b_{1,1},I_{i})}\\}_{i=1}^{r}$ instead. Note
we cannot have $l=m=0$ as ${\Gamma}$ has infinite rank. If $r=0$, we set
$W:=\emptyset$.

Next, we have the following finite collection of loop swaps:

$\displaystyle B:=$ $\displaystyle\\{\alpha_{ij}:=\text{swaps
}a_{i,1}\leftrightarrow a_{j,1}\ |\ 1\leq i<j\leq l\\}$
$\displaystyle\cup\\{\beta_{ij}:=\text{swaps }b_{i,1}\leftrightarrow b_{j,1}\
|\ 1\leq i<j\leq m\\}$ $\displaystyle\cup\\{\gamma_{ij}:=\text{swaps
}a_{i,1}\leftrightarrow b_{j,1}\ |\ 1\leq i\leq l,\ 1\leq j\leq m\\}.$

In words, $B$ is the collection of all loop swaps between loops that are
adjacent to $K$.

Finally, we need a finite collection of loop shifts. The graph ${\Gamma}$ has
only finitely many ends accumulated by loops, so by C,
$H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})$ has finite rank. Let $H$ be
a finite collection of primitive loop shifts dual to a finite basis of
$H^{1}(\operatorname{PMap}({\Gamma});\mathbb{Z})$.

We claim that the set

$\displaystyle\mathcal{S}:=\mathcal{V}_{K}\cup W\cup B\cup H$

is a CB generating set for $\operatorname{PMap}({\Gamma})$. Note that
$\mathcal{S}$ is CB since $\mathcal{V}_{K}$ is CB by 4.5 and each of $W,B,$
and $H$ is simply a finite set. Thus we only need to verify that $\mathcal{S}$
is a generating set for $\operatorname{PMap}({\Gamma})$. We will first check
that $\mathcal{S}$ generates $\operatorname{PMap_{\it c}}({\Gamma})$.

######  Lemma 4.6.

If $K^{\prime}\subset{\Gamma}$ is any connected compact subset of ${\Gamma}$,
then $\operatorname{PMap}(K^{\prime})\subset\langle\mathcal{S}\rangle$.

Before we give the proof of this lemma we review finite generating sets for
$\operatorname{Aut}(F_{n})$. Let $F_{n}$ be a free group of rank $n$, and
denote by $a_{1},\ldots,a_{n}$ its free generators. In 1924, Nielsen [Nie24]
proved a finite presentation of $\operatorname{Aut}(F_{n})$, with the
generating set
$\\{\tau_{i}\\}_{i=1}^{n}\cup\\{\sigma_{ij},\lambda_{ij},\rho_{ij}\\}_{1\leq
i\neq j\leq n}$, where:

$\displaystyle\tau_{i}=\begin{cases}a_{i}\mapsto a_{i}^{-1},\\\ a_{j}\mapsto
a_{j}&\text{for $j\neq i$.}\end{cases}$
$\displaystyle\sigma_{ij}=\begin{cases}a_{i}\leftrightarrow a_{j},\\\
a_{k}\mapsto a_{k}&\text{for $k\neq i,j$.}\end{cases}$
$\displaystyle\lambda_{ij}=\begin{cases}a_{i}\mapsto a_{j}a_{i},\\\
a_{k}\mapsto a_{k}&\text{for $k\neq i,j$.}\end{cases}$
$\displaystyle\rho_{ij}=\begin{cases}a_{i}\mapsto a_{i}a_{j},\\\ a_{k}\mapsto
a_{k}&\text{for $k\neq i,j$.}\end{cases}$

We call $\tau_{i}$ a flip, $\sigma_{ij}$ a transposition, and
$\lambda_{ij},\rho_{ij}$ left/right Nielsen automorphisms respectively. In
fact, Armstrong–Forrest–Vogtmann [AFV08, Theorem 1] reduced this generating
set to consist only of involutions:

($\dagger$)
$\\{\tau_{i}\\}_{i=1}^{n}\cup\\{\sigma_{i,i+1}\\}_{i=1}^{n-1}\cup\\{\tau_{2}\lambda_{12}\\}.$

###### Proof of 4.6.

Let $K^{\prime}$ be a connected compact subset of ${\Gamma}$. Without loss of
generality, we can increase the size of $K^{\prime}$ so that it is as in
Figure 7. In particular, $K^{\prime}$ satisfies the following:

  * •

$K^{\prime}$ contains at least two loops of $L_{1}$ (or $M_{1}$ if $l=0$),

  * •

$K^{\prime}$ contains at least one loop from every monster summand,

  * •

the vertices in $K^{\prime}$ are contained in its interior,

  * •

every component of ${\Gamma}\setminus K^{\prime}$ is infinite,

  * •

$K^{\prime}$ is connected and contains the wedge point $K$.

Note that the last two properties imply that $K^{\prime}$ contains a
subsegment of every ray summand $R_{i}$.

Figure 7. Illustration of $K^{\prime}$ in ${\Gamma}$. Here we have
$l=2,m=3,r=3$ and $\operatorname{PMap}(K^{\prime})\cong
F_{9}^{11}\rtimes\operatorname{Aut}(F_{9})$. We may assume $K^{\prime}$
contains: at least one loop from every monster summand, at least two loops
from one of the monster summands, and initial segments of the ray summands, as
well as $K$. If needed, we can further enlarge $K^{\prime}$ such that it
contains the vertices in the interior, and it only contains the entirety of
the loops.

By 2.3 and 2.4, we have that $\operatorname{PMap}(K^{\prime})\cong
F_{m}^{k}\rtimes\operatorname{Aut}(F_{m})$ for some $k>0$ and
$m=\operatorname{rk}(K^{\prime})$. We first check that
$\langle\mathcal{S}\rangle$ contains an Armstrong–Forrest–Vogtmann generating
set for $\operatorname{Aut}(F_{m})$. Relabel the loops of $K^{\prime}$ by
$a_{1},\ldots,a_{m}$ in the following manner. The loop of $L_{1}$ closest to
$K$ is labeled $a_{1}$, the next loop in $L_{1}$ is $a_{2}$, continued until
the loops of $L_{1}$ are exhausted. Then the next loop, say $a_{j+1}$, is the
first loop on $L_{2}$ etc., until all of the loops in all of the $L_{i}$ are
exhausted. Finally, continue relabeling by $a_{\bullet}$’s the loops in
$M_{1}$ through $M_{m}$, in the same process. Note that when $l=0$, then
$a_{1}$ and $a_{2}$ are contained in $M_{1}$.

Note that we immediately have
$\tau_{1},\ldots,\tau_{m},\lambda_{12}\in\mathcal{V}_{K}\subset\mathcal{S}$.
Therefore it remains to check that
$\sigma_{i,i+1}\in\langle\mathcal{S}\rangle$ for all $i=1,\ldots,m-1$. Each
such $\sigma_{i,i+1}$ either swaps two adjacent loops in a single component of
$K^{\prime}\setminus K$ or swaps the last loop in a component of
$K^{\prime}\setminus K$ with the first loop in the next component. In the
former case we already have that $\sigma_{i,i+1}\in\mathcal{V}_{K}$. For the
latter case, let $a_{t}$ be the first loop in the component of
$K^{\prime}\setminus K$ containing $a_{i}$. Then consider the loop swap
$\sigma_{i,t}$ that swaps $a_{i}$ with $a_{t}$ (note those two loops could
coincide, and then $\sigma_{i,t}$ is the identity) and let $\sigma_{t,i+1}$ be
the loop swap that swaps $a_{t}$ with $a_{i+1}$, which is the first loop in
the component of $K^{\prime}\setminus K$ containing $a_{i+1}$. Then we have
that $\sigma_{i,1}\in\mathcal{V}_{K}$, $\sigma_{t,i+1}\in B$ and
$\sigma_{i,i+1}=\sigma_{i,t}\sigma_{t,i+1}\sigma_{i,t}\in\langle\mathcal{S}\rangle$.
Thus we see that every Armstrong–Forrest–Vogtmann generator for the
$\operatorname{Aut}(F_{m})$ subgroup of $\operatorname{PMap}(K^{\prime})\cong
F_{m}^{k}\rtimes\operatorname{Aut}(F_{m})$ is contained in
$\langle\mathcal{S}\rangle$.

Finally we need to be able to obtain each of the $k$ factors of $F_{m}$ in
$\operatorname{PMap}(K^{\prime})$. Each $F_{m}$ subgroup can be identified
with the subgroup of the collection of word maps on an interval adjacent to
the boundary of $K^{\prime}$. Recall by 2.3 there are $k+1$ such boundary
adjacent intervals, so say $I_{1},\ldots,I_{k+1}$. Since we have already
generated the $\operatorname{Aut}(F_{m})$ subgroup of
$\operatorname{PMap}(K^{\prime})$ with $\mathcal{S}$ and we can change the
word of the word map using 2.12 and 2.13, it suffices to show that a _single_
word map on each interval $I_{j}$ that maps onto a generator of $F_{m}$ is in
$\langle\mathcal{S}\rangle$. However, we even have one such word map in
$\mathcal{S}$ already. Indeed, if $I_{j}$ is contained in some ray then we
have already added a corresponding word map to $W$. Otherwise, if $I_{j}$ is
contained in some monster summand, then there is an appropriate word map
already in $\mathcal{V}_{K}$ obtained by mapping $I_{j}$ over the first loop
of that summand. We can thus conclude that
$\operatorname{PMap}(K^{\prime})\cong
F_{m}^{k}\rtimes\operatorname{Aut}(F_{m})$ in contained in
$\langle\mathcal{S}\rangle$. ∎

We are now ready to prove A. Note that in the above lemma we never made use of
the loop shifts in $H$. They will now be used to push any random mapping class
into the closure of the compactly supported mapping classes.

###### Proof of A.

As discussed in the beginning of the section, the only if direction comes from
4.2 and 4.3. Now we prove the if direction. When ${\Gamma}$ is a tree, we have
$\operatorname{PMap}({\Gamma})=1$ by 2.6. If ${\Gamma}$ has finite rank,
$\operatorname{PMap}({\Gamma})$ is finitely generated by 4.4. Also if
${\Gamma}$ is either the Loch Ness monster or Millipede monster graph, then by
2.16, $\operatorname{PMap}({\Gamma})$ is CB. Hence we may assume
$1\leq|E_{\ell}|<\infty$, there is no accumulation point in $E\setminus
E_{\ell}$, and ${\Gamma}$ is neither the Loch Ness monster nor the Millipede
monster graphs.

Let $\mathcal{S}$ be as defined above; $\mathcal{S}=\mathcal{V}_{K}\cup W\cup
B\cup H$. We will show that $\mathcal{S}$ generates
$\operatorname{PMap}({\Gamma})$. Let $f\in\operatorname{PMap}({\Gamma})$. If
$|E_{\ell}|=1$, then
$\operatorname{PMap}({\Gamma})=\overline{\operatorname{PMap_{\it
c}}({\Gamma})}$ by B, so we obtain $f\in\overline{\operatorname{PMap_{\it
c}}({\Gamma})}$. Otherwise, if $|E_{\ell}|\geq 2$, then by postcomposing $f$
with primitive loop shifts in $H$, we may assume the flux of $f$ is zero with
respect to any 2-partition of $E_{\ell}$. By 3.11, we can assume
$f\in\overline{\operatorname{PMap_{\it c}}({\Gamma})}$ for this case as well.

Then there exists a compact set $K^{\prime}$ containing $K$, and
$g\in\operatorname{PMap_{\it c}}({\Gamma})$ such that $g$ is totally supported
in $K^{\prime}$ and $fg^{-1}\in\mathcal{V}_{K}$. Therefore, it suffices to
show that $g$ is contained in the group generated by $\mathcal{S}$. Since $g$
is totally supported in $K^{\prime}$, the map $g$ can be identified with an
element in $\operatorname{PMap}(K^{\prime})$, which is contained in
$\langle\mathcal{S}\rangle$ by 4.6. This concludes the proof that
$\operatorname{PMap}({\Gamma})$ is generated by $\mathcal{S}$. Finally,
$\mathcal{S}$ is CB as it is the union of three finite sets $W,B$, and $H$ and
the set $\mathcal{V}_{K}$, which is CB by 4.5. ∎

##  5\. Residual finiteness

In this section, we prove D:

######  Theorem 5.1 (D, revisited).

$\operatorname{PMap}({\Gamma})$ is residually finite if and only if ${\Gamma}$
has a finite rank.

###  5.1. Forgetful map

Throughout this section, we let ${\Gamma}$ be a locally finite, infinite graph
with no ends accumulated by loops. That is, $E_{\ell}({\Gamma})=\emptyset$ but
$E({\Gamma})\neq\emptyset$. Fix an end $\alpha_{0}\in E({\Gamma})$. Define
$E_{<\infty}({\Gamma})$ as the collection of finite subsets of $E({\Gamma})$
containing $\alpha_{0}$:

$E_{<\infty}({\Gamma})=\\{\mathcal{E}\subset
E({\Gamma}):\alpha_{0}\in\mathcal{E},\text{ and }|\mathcal{E}|<\infty\\}.$

For each $\mathcal{E}\in E_{<\infty}({\Gamma})$, we define the graph
${\Gamma}_{\mathcal{E}}$ as a subgraph of ${\Gamma}$ such that:

  * •

$\operatorname{rk}{\Gamma}_{\mathcal{E}}=\operatorname{rk}{\Gamma}$, and

  * •

$E({\Gamma}_{\mathcal{E}})=\mathcal{E}$.

Note ${\Gamma}_{\mathcal{E}}$ is properly homotopic to the core graph
${\Gamma}_{c}$ of ${\Gamma}$ with $|\mathcal{E}|$ rays attached.

Now we use 2.3:
$\operatorname{PMap}({\Gamma})\cong\mathcal{R}\rtimes\operatorname{PMap}({\Gamma}_{c}^{*})$
if $E({\Gamma})\setminus E_{\ell}({\Gamma})$ is nonempty and compact. In our
case ${\Gamma}$ is of infinite type and has no ends accumulated by loops, so
$E({\Gamma})\setminus E_{\ell}({\Gamma})=E({\Gamma})$ is automatically
nonempty and compact. Now we denote by $\mathcal{R}_{\mathcal{E}}$ the
$\mathcal{R}$ subgroup for ${\Gamma}_{\mathcal{E}}$. Then we have a map
$\rho_{\mathcal{E}}:\mathcal{R}\to\mathcal{R}_{\mathcal{E}}$ by ‘restricting’
the domain to $E({\Gamma}_{\mathcal{E}})$. Namely, given a locally constant
map $[f:E({\Gamma})\to\pi_{1}({\Gamma},\alpha_{0})]\in\mathcal{R}$, we define
$\rho_{\mathcal{E}}(f):=f|_{E({\Gamma}_{\mathcal{E}})}$, where we note that
$f|_{E({\Gamma}_{\mathcal{E}})}:E({\Gamma}_{\mathcal{E}})\to\pi_{1}({\Gamma},\alpha_{0})=\pi_{1}({\Gamma}_{\mathcal{E}},\alpha_{0})$
is a locally constant map to $\pi_{1}({\Gamma}_{\mathcal{E}},\alpha_{0})$, so
$\rho_{\mathcal{E}}(f)\in\mathcal{R}_{\mathcal{E}}$.

######  Lemma 5.2.

The restriction map
$\rho_{\mathcal{E}}:\mathcal{R}\to\mathcal{R}_{\mathcal{E}}$ is a group
homomorphism.

###### Proof.

Recall the group operation in $\mathcal{R}$ is the pointwise multiplication in
$\pi_{1}({\Gamma},\alpha_{0})$. Hence the restriction on $f\cdot g$ for
$f,g\in\mathcal{R}$ is just the product of their restrictions:

$\rho_{\mathcal{E}}(f\cdot g)=(f\cdot
g)|_{E({\Gamma}_{\mathcal{E}})}=f|_{E({\Gamma}_{\mathcal{E}})}\cdot
g|_{E({\Gamma}_{\mathcal{E}})}=\rho_{\mathcal{E}}(f)\cdot\rho_{\mathcal{E}}(g).\qed$

Observe ${\Gamma}_{c}^{*}=({\Gamma}_{\mathcal{E}})_{c}^{*}$. Hence, from the
lemma we can build the homomorphism $\mathcal{F}_{\mathcal{E}}$ on
$\operatorname{PMap}({\Gamma})$ to
$\operatorname{PMap}({\Gamma}_{\mathcal{E}})$ by just letting
$\mathcal{F}_{\mathcal{E}}=\rho_{\mathcal{E}}\times\operatorname{Id}$ on the
decomposition
$\operatorname{PMap}({\Gamma})=\mathcal{R}\rtimes\operatorname{PMap}({\Gamma}_{c}^{*})$
given in 2.3:

$\mathcal{F}_{\mathcal{E}}:\operatorname{PMap}({\Gamma})\to\operatorname{PMap}({\Gamma}_{\mathcal{E}}),$

which we will call as the forgetful homomorphism to $\mathcal{E}\subset
E({\Gamma})$.

###  5.2. Finite rank if and only if residually finite

Now we prove D. The following lemma is used for the only if direction of the
proof. Denote by $\operatorname{SAut}(F_{n})$ the unique index 2 subgroup of
$\operatorname{Aut}(F_{n})$.

######  Fact 5.3 ([BKP19, Theorem 9.1]).

There exists a strictly increasing sequence of integers $\\{a_{n}\\}_{n\geq
3}$ such that for $n\geq 3$, every nontrivial finite quotient of
$\operatorname{SAut}(F_{n})$ has cardinality of at least $a_{n}$.

###### Proof of D.

Suppose that ${\Gamma}$ has finite rank $n$. If ${\Gamma}$ has no ends then
$\operatorname{PMap}({\Gamma})$ is isomorphic to $\operatorname{Out}(F_{n})$,
which is residually finite by [Gro75]. If ${\Gamma}$ has only one end, then
$\operatorname{PMap}({\Gamma})$ is isomorphic to $\operatorname{Aut}(F_{n})$,
which is residually finite by [Bau63]. If ${\Gamma}$ has finitely many ends,
then $\operatorname{PMap}({\Gamma})\cong
F_{n}^{{|E|-1}}\rtimes\operatorname{Aut}(F_{n})$ which is again residually
finite as both factors are residually finite and $F_{n}^{|E|-1}$ is finitely
generated [Mal58, Theorem 1].

Now we assume ${\Gamma}$ has finite rank and infinitely many ends. The proof
is similar to the proof for infinite-type surfaces; [PV18, Proposition 4.6].
Let $f\in\operatorname{PMap}({\Gamma})$ be a nontrivial element. Since
${\Gamma}$ is of finite rank and $f$ is proper, it follows that
$({\Gamma}\setminus{\Gamma}_{c})\cap\operatorname{supp}(f)$ is compact. In
particular, there exists some finite set $\mathcal{E}\subset E$ such that
$({\Gamma}_{\mathcal{E}}\setminus{\Gamma}_{c})\cap\operatorname{supp}(f)$ is
still nonempty. This implies that the forgetful map
$\mathcal{F}_{\mathcal{E}}:\operatorname{PMap}({\Gamma})\to\operatorname{PMap}({\Gamma}_{\mathcal{E}})$
sends $f$ to a nontrivial element
$\mathcal{F}_{\mathcal{E}}(f)\in\operatorname{PMap}({\Gamma}_{\mathcal{E}})$.
However, we know that $\mathcal{E}$ has finite end space so
$\operatorname{PMap}({\Gamma}_{\mathcal{E}})$ is residually finite by the
previous paragraph. Therefore, there exists a homomorphism
$\psi:\operatorname{PMap}({\Gamma}_{\mathcal{E}})\to F$ for some finite group
$F$ so that $\psi(\mathcal{F}_{\mathcal{E}}(f))$ is nontrivial. Thus
$\operatorname{PMap}({\Gamma})$ is residually finite.

Conversely, let ${\Gamma}$ have infinite rank and assume it is in standard
form. Let $\\{{\Gamma}_{k}\\}$ be a compact exhaustion of ${\Gamma}$ by
connected subgraphs. Then there exist non-decreasing sequences
$\\{n_{k}\\},\\{e_{k}\\}$ such that $\operatorname{PMap}({\Gamma}_{k})\cong
F_{n_{k}}^{e_{k}}\rtimes\operatorname{Aut}(F_{n_{k}})$. Here $e_{k}+1$ is the
number of boundaries of ${\Gamma}_{k}$ (i.e. the size of
$\overline{{\Gamma}\setminus{\Gamma}_{k}}\cap{\Gamma}_{k}$), and $n_{k}$ is
the rank of ${\Gamma}_{k}$. As ${\Gamma}$ has infinite rank, we have
$\lim_{k\to\infty}n_{k}=\infty.$ Also, note $\operatorname{Aut}(F_{n_{k}})$
has the unique index 2 subgroup $\operatorname{SAut}(F_{n_{k}})$ for each $k$,
whose isomorphic copy in $\operatorname{PMap}({\Gamma}_{k})$ we denote by
$G_{k}$. The group $\operatorname{Aut}(F_{n_{k}})$ can be identified with the
subgroup of mapping classes totally supported on the core graph of
${\Gamma}_{k}$, and $G_{k}\cong\operatorname{SAut}(F_{n_{k}})$ with the set of
those mapping classes that preserve orientation. Since the core graph of
${\Gamma}_{k}$ is contained in the core graph of ${\Gamma}_{k+1}$, and
orientation preserving mapping classes on ${\Gamma}_{k}$ are orientation
preserving on ${\Gamma}_{k+1}$, it follows that we have the inclusion
$G_{k}\hookrightarrow G_{k+1}$. Hence the direct limit
$\operatorname{SAut}_{\infty}({\Gamma}):=\varinjlim G_{k}$ is a well-defined
subgroup of $\operatorname{PMap}({\Gamma})$.

We claim that $\operatorname{SAut}_{\infty}({\Gamma})$ has no finite
quotients. To do this, suppose $H$ is a proper normal subgroup of
$\operatorname{SAut}_{\infty}({\Gamma})$ with finite index $r\geq 2$. Then as
$H$ is a proper subgroup of $\operatorname{SAut}_{\infty}({\Gamma})$ and
$\varinjlim G_{k}=\operatorname{SAut}_{\infty}({\Gamma})$, it follows that
there exists some $k_{0}$ such that whenever $k\geq k_{0}$, $G_{k}$ is not
contained in $H$. Hence $H\cap G_{k}$ is a _proper_ normal subgroup of
$G_{k}$. Note

$1\neq[G_{k}:G_{k}\cap H]\leq[\operatorname{SAut}_{\infty}({\Gamma}):H]=r,$

but the minimal finite index of proper subgroups of
$G_{k}\cong\operatorname{SAut}(F_{n_{k}})$ increases as $k$ does by 5.3.
Therefore, $[G_{k}:G_{k}\cap H]$ cannot be uniformly bounded by $r$; giving a
contradiction. Therefore $\operatorname{SAut}_{\infty}({\Gamma})$ has no
nontrivial finite quotient, implying that both $\operatorname{PMap}({\Gamma})$
and $\operatorname{Map}({\Gamma})$ are not residually finite. ∎

##  6\. Tits alternative

In a series of three papers [BFH00, BFH05, BFH04], Bestvina, Feighn, and
Handel prove that $\operatorname{Out}(F_{n})$ satisfies what we call the
strong Tits alternative: Every subgroup either contains a nonabelian free
group or is virtually abelian. The same was previously known for mapping class
groups of compact surfaces by work of Ivanov, McCarthy, and
Birman–Lubotzky–McCarthy [Iva84, McC85, BLM83]. However, it was shown by
Lanier and Loving [LL20] that this is not the case for big mapping class
groups. They prove that big mapping class groups _never_ satisfy the strong
Tits alternative by showing that they always contain a subgroup isomorphic to
the wreath product $\mathbb{Z}\wr\mathbb{Z}$. In [AHL+21], the authors extend
this idea to find many subgroups isomorphic to wreath products. Allcock
[All21] further showed that most big mapping class groups fail the (standard)
Tits alternative by finding a poison subgroup that surjects onto a Grigorchuck
group. A groups satisfies the Tits alternative if every subgroup either
contains a nonabelian subgroup or is virtually solvable. Note that some
references require subgroups be finitely generated, but we do not need to make
that restriction.

###  6.1. Infinite rank: Fails to satisfy TA

In this section, we find poison subgroups (analogous to the surface case) in
$\operatorname{PMap}({\Gamma})$ for graphs ${\Gamma}$ of infinite rank.

######  Theorem 6.1.

Let ${\Gamma}$ be a locally finite graph of infinite rank. Then
$\operatorname{PMap}({\Gamma})$ contains a subgroup isomorphic to
$\operatorname{Aut}(F_{n})\wr\mathbb{Z}$ for all $n\in\mathbb{N}$.

###### Proof.

Recall that to define the wreath product $G\wr\mathbb{Z}$, we need a
$\mathbb{Z}$-indexed set of copies of $G$, denoted by
$\\{G_{i}\\}_{i\in\mathbb{Z}}$. Then $\mathbb{Z}$ acts on the index set by
translation, so it also acts on $\oplus_{i\in\mathbb{Z}}G_{i}$ by translation
on the indices. Now set $G=\operatorname{Aut}(F_{n})$ and denote by $\phi$ the
translation action by $\mathbb{Z}$ on the direct sum. We then define

$\operatorname{Aut}(F_{n})\wr\mathbb{Z}:=\left(\bigoplus_{\mathbb{Z}}\operatorname{Aut}(F_{n})\right)\rtimes_{\phi}\mathbb{Z}.$

To realize this group as a subgroup of $\operatorname{PMap}({\Gamma})$, we
will find $\mathbb{Z}$ copies of $\operatorname{Aut}(F_{n})$ together with a
translation action.

For each $n\in\mathbb{N}$, let $\Delta_{n}$ be the graph obtained from a line
identified with $\mathbb{R}$ with a wedge of $n$ circles attached by an edge
at each integer point; see Figure 8. If ${\Gamma}$ has at least two ends
accumulated by loops, we can properly homotope ${\Gamma}$ to have $\Delta_{n}$
as a subgraph.

Figure 8. The graph $\Delta_{4}$, with two ends accumulated by roses with $4$
petals. It admits a translation of roses, denoted by the green dotted arrow.
Up to proper homotopy, such graph arises as a subgraph of any graph with at
least two ends accumulated by loops.

For each $i\in\mathbb{Z}$, let $R_{i}$ be the wedge of circles supported above
the integer point $i$ in $\Delta_{n}\subset\Gamma$. Let $G_{i}$ be the
subgroup of elements of $\operatorname{PMap}({\Gamma})$ which are totally
supported on $R_{i}$. Each $G_{i}$ is isomorphic to
$\operatorname{Aut}(F_{n})$ (see 2.4) and the $G_{i}$’s have disjoint total
support, so $\langle
G_{i}\rangle_{i\in\mathbb{Z}}\cong\oplus_{\mathbb{Z}}\operatorname{Aut}(F_{n})$.
There is a shift map, call it $h$, that translates along $\Delta_{n}$ by $+1$
on the line and maps $R_{i}$ to $R_{i+1}$ isometrically. Because
$h^{m}G_{i}=G_{i+m}h^{m}$, the subgroup of $\operatorname{PMap}({\Gamma})$
generated by $G_{0}$ and $h$ is isomorphic to
$\operatorname{Aut}(F_{n})\wr\mathbb{Z}$.

In general, if ${\Gamma}$ has at least one end accumulated by loops, we can
embed a copy of $\Delta_{n}$ into ${\Gamma}$ where the images of the two ends
of $\Delta_{n}$ are not distinct. The corresponding “shift map” will no longer
be shifting between distinct ends, but this does not affect the construction
of an $\operatorname{Aut}(F_{n})\wr\mathbb{Z}$ subgroup. ∎

6.1 immediately tells us that $\operatorname{PMap}({\Gamma})$ fails the strong
Tits alternative because $\mathbb{Z}\wr\mathbb{Z}$ is a subgroup of
$\operatorname{Aut}(F_{n})\wr\mathbb{Z}$. In [All21], Allcock shows that big
mapping class groups of surfaces with infinite genus fail the (standard) Tits
alternative. His idea is to find elements of the mapping class group that
“look like” the action of the Grigorchuck group on a rooted binary tree.
Because these elements are not of finite order, the resulting subgroup of the
mapping class group is an extension of the Grigorchuck group. When this same
idea is implemented in the pure mapping class group of a graph, we instead
find an exact copy of Grigorchuck’s group. Many graphs, such as an infinite
binary tree, also contain Grigorchuck’s group as a subgroup of their _full_
mapping class group in the obvious way.

######  Theorem 6.2.

Let ${\Gamma}$ be a locally finite graph of infinite rank. Then
$\operatorname{PMap}({\Gamma})$ contains a subgroup isomorphic to the
Grigorchuck group.

###### Proof.

First, we define the proper homotopy equivalences called $a,b,c,d$ on an
infinite binary tree $T$ as in Figure 9. Note only $a$ swaps the level 1
branches. Each of the other three homotopy equivalences $b,c,d$ misses
$(3k+1),3k,(3k-1)$-st branch swaps for $k\geq 1$ respectively, as well as the
level-1 swap. These four elements generate the Grigorchuck group, $G$ [Gri80,
GP08].

Figure 9. Proper homotopy equivalences $a,b,c$ and $d$ on infinite binary tree
$T$. Each green arrow denotes the swap of the two subtrees induced by the swap
of the two branches.

Now let $\Delta$ be the infinite graph with one end accumulated by loops,
constructed as in Figure 10. Specifically, we start with a ray and label the
countably many vertices by $v_{1},v_{2},\ldots$ etc. Attach a finite binary
tree $T_{i}$ of level $i$ to $v_{i}$ for each $i\geq 1$. Then attach a single
loop at each leaf of the trees. For any graph ${\Gamma}$ with infinite rank,
we can apply a proper homotopy equivalence so that $\Delta$ is a subgraph.
Hence, $\operatorname{PMap}(\Delta)\leq\operatorname{PMap}({\Gamma})$, so it
suffices to find a Grigorchuck group inside $\operatorname{PMap}(\Delta)$.

Figure 10. The graph $\Delta$ and the proper homotopy equivalence $\hat{b}$ on
$T_{1},T_{2},\ldots\subset\Delta$ mimicking the definition of $b$ on $T$.

Define a proper homotopy equivalence $\hat{b}$ as the map on _finite_ binary
trees $T_{1},T_{2},\ldots$ by ‘mimicking’ $b$ defined on the _infinite_ binary
tree $T$. See Figure 10 for an illustration of $\hat{b}$, denoted in green
arrows. Similarly define $\hat{a},\hat{c}$ and $\hat{d}$ from $a,c$ and $d$.
Denote by $\widetilde{a},\widetilde{b},\widetilde{c}$ and $\widetilde{d}$ the
proper homotopy classes of $\hat{a},\hat{b},\hat{c}$ and $\hat{d}$
respectively. Following the same proof of [All21, Lemma 4.1], we see that
$\widetilde{a},\widetilde{b},\widetilde{c},\widetilde{d}$ satisfy exactly the
defining relations of $G$, and
$\widetilde{G}:=\langle\widetilde{a},\widetilde{b},\widetilde{c},\widetilde{d}\rangle$
is isomorphic to the Grigorchuck group. ∎

######  Corollary 6.3.

Let ${\Gamma}$ be a locally finite graph of infinite rank. Then
$\operatorname{PMap}({\Gamma})$ and $\operatorname{Map}({\Gamma})$ fail the
Tits alternative.

###  6.2. Finite rank: Satisfies TA

On the other hand, when ${\Gamma}$ has finite rank, we get the following
contrasting result.

######  Theorem 6.4.

Let ${\Gamma}$ be a locally finite graph with finite rank. Then
$\operatorname{PMap}({\Gamma})$ satisfies the Tits alternative. That is, every
finitely generated subgroup is either virtually solvable or contains a free
group.

We first need the following stability property of the Tits alternative.

######  Proposition 6.5.

Satisfying the Tits alternative is stable under subgroups, finite-index
supergroups, and group extensions. More precisely,

  1. (1)

Let $H\leq G$. If $G$ satisfies the Tits alternative, then so does $H$.

  2. (2)

Let $H\leq G$ with $[G:H]<\infty$. If $H$ satisfies the Tits alternative, then
so does $G$.

  3. (3)

(cf. [Can11, Proposition 6.3]) Suppose the groups $K,G,H$ form a short exact
sequence as follows:

$1\longrightarrow K\longrightarrow G\longrightarrow H\longrightarrow 1.$

If $K$ and $H$ satisfy the Tits alternative, then so does $G$.

###### Proof.

Claim (1) holds because every subgroup of $H$ is a subgroup of $G$. Claim (2)
will follow from (3), as finite groups are virtually trivial so satisfy the
Tits alternative.

Now we prove (3). Let $L\leq G$ be a subgroup. Then we have the following
commutative diagram:

${1}$${K}$${G}$${H}$${1}$${1}$${K\cap
L}$${L}$${q(L)}$${1}$$\scriptstyle{q}$$\scriptstyle{q}$

Indeed, $K\cap L\trianglelefteq L$ and $q(L)\cong L/(K\cap L)\leq H$. By (1),
both $K\cap L$ and $q(L)$ satisfy Tits alternative. If $K\cap L$ has $F_{2}$
as a subgroup, then $L$ has $F_{2}$ as a subgroup. If $q(L)$ has $F_{2}$ has a
subgroup, then we can find a section of $q$ to lift $F_{2}$ inside $L$. Hence,
we may assume both $K\cap L$ and $q(L)$ are virtually solvable. In this case,
the following fact finishes the proof.

######  Fact 6.6 ([Din12, Lemma 5.5], see also [Can11, Lemme 6.1]).

Suppose $N$ is a normal subgroup of a group $G$. If both $N$ and $G/N$ are
virtually solvable, then $G$ is virtually solvable.

Hence $L$ is virtually solvable, concluding that $G$ satisfies Tits
alternative. ∎

Now we are ready to prove 6.4.

###### Proof of 6.4.

Let $\operatorname{rk}{\Gamma}=n.$ Then we have the following short exact
sequence [AB21, Theorem 3.5]:

$1\longrightarrow\mathcal{R}\longrightarrow\operatorname{PMap}({\Gamma})\longrightarrow\operatorname{Aut}(F_{n})\longrightarrow
1,$

where $\mathcal{R}$ is the group of locally constant functions from $E$ to
$F_{n}$ with pointwise multiplication.

The subgroup of $\operatorname{Out}(F_{n+1})$ fixing one $\mathbb{Z}$ factor
is naturally isomorphic to $\operatorname{Aut}(F_{n})$. Recall that
$\operatorname{Out}(F_{n+1})$ satisfies the Tits alternative by [BFH00], so
$\operatorname{Aut}(F_{n})$ does too. We will show that $\mathcal{R}$
satisfies the (strong) Tits alternative, then 6.5 part (3), guarantees that
$\operatorname{PMap}({\Gamma})$ satisfies the Tits alternative as well.

######  Claim.

$\mathcal{R}$ satisfies the strong Tits alternative.

Consider a (not necessarily finitely generated) subgroup
$H\subset\mathcal{R}$. Suppose there exist $\phi,\psi\in\mathcal{R}$ that do
not commute; so there exists an $e\in E$ such that
$\phi(e)\psi(e)\neq\psi(e)\phi(e)$. Now we use the Ping Pong lemma to prove
$\langle\phi,\psi\rangle\cong F_{2}$, which will conclude that $\mathcal{R}$
satisfies the strong Tits alternative. Let $X_{\phi(e)}$ and $X_{\psi(e)}$ as
the set of words in $F_{n}$ that start with $\phi(e)$ and $\psi(e)$
respectively. We note $X_{\phi(e)}$ and $X_{\psi(e)}$ are disjoint as
otherwise $\phi(e)=\psi(e)$, contradicting the assumption
$\phi(e)\psi(e)\neq\psi(e)\phi(e)$. We consider the action of $H$ on $F_{n}$
as:

$\phi\cdot w:=\phi(e)w,\qquad\text{for $\phi\in\mathcal{R}$ and $w\in
F_{n}$}.$

Then the same assumption $\phi(e)\psi(e)\neq\psi(e)\phi(e)$ implies that
$\phi\cdot X_{\psi(e)}\subset X_{\phi(e)}$ and $\psi\cdot X_{\phi(e)}\subset
X_{\psi(e)}$. Therefore, by the Ping-Pong lemma, we conclude
$\langle\phi,\psi\rangle\cong F_{2}$ so $\mathcal{R}$ satisfies the (strong)
Tits alternative. ∎

We now extend these results to determine which full mapping class groups
satisfy the Tits alternative.

######  Corollary 6.7.

Let ${\Gamma}$ be a locally finite, infinite graph. Then
$\operatorname{Map}({\Gamma})$ satisfies the Tits alternative if and only if
${\Gamma}$ has finite rank and finite end space.

###### Proof.

We divide into cases. First, if ${\Gamma}$ has at least one end accumulated by
loops, then $\operatorname{Map}({\Gamma})$ fails the Tits alternative by 6.3.
Otherwise, ${\Gamma}$ has finite rank, and we continue to divide into cases.
If ${\Gamma}$ has finite end space, then $\operatorname{Map}({\Gamma})$ is a
finite extension of $\operatorname{PMap}({\Gamma})$, so by 6.5 property (2),
the full mapping class group $\operatorname{Map}({\Gamma})$ satisfies the Tits
alternative. If ${\Gamma}$ has countable end space, then we can modify the
proof of 6.2 by replacing the loops with rays, to realize Grigorchuck’s group
as a subgroup of $\operatorname{Map}({\Gamma})$. If the end space of
${\Gamma}$ is uncountable, then there is a closed subset of the ends which is
homeomorphic to the whole Cantor set, so contains Grigorchuck’s group in the
natural way, and again $\operatorname{Map}({\Gamma})$ fails the Tits
alternative. ∎

The strong Tits alternative is not stable under group extensions (consider
$\mathbb{Z}\wr\mathbb{Z}$). So, the best we could conclude about
$\operatorname{PMap}({\Gamma})$ from the decomposition as
$\mathcal{R}\rtimes\operatorname{Aut}(F_{n})$ was 6.4. However, our proof that
$\mathcal{R}$ satisfies the strong Tits alternative actually shows the
slightly stronger statement: Any two elements of $\mathcal{R}$ which do not
commute generate $F_{2}$. This property could be useful in answering the
question,

######  Question 6.8.

If ${\Gamma}$ is a locally finite graph of finite rank, does
$\operatorname{PMap}({\Gamma})$ satisfy the strong Tits alternative?

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