# Group actions on multitrees and the
$K$-theory of their crossed products

Nathan Brownlowe  ,  Jack Spielberg  ,  Anne Thomas  and  Victor Wu  Nathan
Brownlowe, Anne Thomas and Victor Wu
School of Mathematics and Statistics
The University of Sydney<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Jack Spielberg, School of
Mathematical and Statistical Sciences
Arizona State University
USA<EMAIL_ADDRESS>
###### Abstract.

We study group actions on multitrees, which are directed graphs in which there
is at most one directed path between any two vertices. In our main result we
describe a six-term exact sequence in $K$-theory for the reduced crossed
product $C_{0}(\partial E)\rtimes_{r}G$ induced from the action of a countable
discrete group $G$ on a row-finite, finitely-aligned multitree $E$ with no
sources. We provide formulas for the $K$-theory of $C_{0}(\partial
E)\rtimes_{r}G$ in the case where $G$ acts freely on $E$, and in the case
where all vertex stabilisers are infinite cyclic. We study the action
$G\curvearrowright\partial E$ in a range of settings, and describe minimality,
local contractivity, topological freeness, and amenability in terms of
properties of the underlying data. In an application of our main theorem, we
describe a six-term exact sequence in $K$-theory for the crossed product
induced from a group acting on the boundary of an undirected tree.

###### 2010 Mathematics Subject Classification:

##  1\. Introduction

Group actions on (undirected) trees have for many years provided interesting
objects of study from the point of view of $C^{*}$-algebras. While the groups
themselves give important case studies in $C^{*}$-simplicity [12] and
amenability [14], the action is best studied via a crossed product
$C^{*}$-algebra. While crossed products by groups acting on trees have been
studied in general [20], the induced action of the group on the boundary of
the tree on which it acts has been particulary fertile ground for crossed
products, and examples such as the Cuntz–Krieger algebras of [24, 22], and the
Bunce–Deddens algebras of [19], have benefited from a crossed-product model
arising from such boundary actions.

Group actions on trees are also central to the field of geometric group
theory. Here, they are studied using Bass–Serre theory, which establishes a
bijection between group actions on trees and quotient structures called graphs
of groups [23, 5]. In [8], the authors established a $C^{*}$-algebraic
Bass–Serre theory by producing a graph-of-groups $C^{*}$-algebra which is
Morita equivalent to the crossed product induced from the corresponding
boundary action of the tree. Using this description, the authors were able to
study free group actions on trees, and group actions with infinite cyclic
stabiliser groups. The work in [8] follows earlier work on $C^{*}$-algebras
associated to graphs of groups by Okayasu [18] and undirected graphs (graphs
of trivial groups) by Cornelissen, Lorscheid and Marcolli [10]. Following [8],
Mundey and Rennie examined the $K$-theory of graph-of-groups $C^{*}$-algebras
by first producing a Cuntz–Pimsner model for graph-of-groups $C^{*}$-algebras
[17].

In this paper we study a more general type of action. A multitree is a
directed graph $E$ in which there is at most one directed path between any two
vertices; these objects were introduced in [13], and they have been studied in
fields relating to computer science, such as data representation and
complexity theory. We consider actions of a countable discrete group $G$ on a
row-finite, finitely-aligned multitree $E$ which has no sources. Such an
action $G\curvearrowright E$ induces an action of $G$ on the boundary
$\partial E$, which is the totally disconnected locally compact Hausdorff
space of shift-tail equivalent infinite paths in $E$. In this article we are
interested in the problem of calculating the $K$-theory of the reduced crossed
product $C_{0}(\partial E)\rtimes_{r}G$.

Our approach to this problem starts with the observation that under certain
assumptions (such as $G$ being exact, or the action of $G$ on
$E^{0}\cup\partial E$ being amenable), we have a short exact sequence

${0}$${C_{0}(E^{0})\rtimes_{r}G}$${C_{0}(E^{0}\cup\partial
E)\rtimes_{r}G}$${C_{0}(\partial E)\rtimes_{r}G}$${0}$$\scriptstyle{\iota}$

giving rise to the six-term exact sequence in $K$-theory

${K_{0}(C_{0}(E^{0})\rtimes_{r}G)}$${K_{0}(C_{0}(E^{0}\cup\partial
E)\rtimes_{r}G)}$${K_{0}(C_{0}(\partial
E)\rtimes_{r}G)}$${K_{1}(C_{0}(\partial
E)\rtimes_{r}G)}$${K_{1}(C_{0}(E^{0}\cup\partial
E)\rtimes_{r}G)}$${K_{1}(C_{0}(E^{0})\rtimes_{r}G).}$$\scriptstyle{\iota_{*,0}}$$\scriptstyle{\iota_{*,1}}$

We then turn our attention to the $K$-groups $K_{*}(C_{0}(E^{0})\rtimes_{r}G)$
and $K_{*}(C_{0}(E^{0}\cup\partial E)\rtimes_{r}G)$. For this we use the
results of Cuntz, Echterhoff and Li [11], in which they prove that if $G$ is a
locally compact group acting on a totally disconnected locally compact space
$\Omega$, and there is a $G$-invariant collection $\mathcal{V}=\\{V_{i}:i\in
I\\}$ of compact open subsets of $\Omega$ satisfying certain properties, then
the $K$-theory of the reduced crossed product $C_{0}(\Omega)\rtimes_{r}G$
decomposes as a direct sum of the $K$-theories of reduced group
$C^{*}$-algebras

(1.1) $K_{*}(C_{0}(\Omega)\rtimes_{r}G)\cong\bigoplus_{[i]\in G\setminus
I}K_{*}(C^{*}_{r}(G_{i})).$

In [11] it is assumed that $\mathcal{V}$ is closed under intersections, which
causes a problem in the setting of multitrees. The collection of cylinder sets
$\\{Z(v):v\in E^{0}\\}$ forms a natural collection of compact open sets in
$\Omega:=E^{0}\cup\partial E$ which is not closed under intersections, but is
rather finitely aligned, in the sense that the intersection $Z(v)\cap Z(w)$ is
a finite disjoint union of other cylinder sets. To overcome this problem we
generalise (in the setting of discrete group actions) the results of [11] to
the setting of finitely-aligned $\mathcal{V}$, and in particular, we show that
the decomposition in Equation (1.1) still holds in this more general setting.
During the preparation of this paper, we learnt that Alistair Miller also
generalised [11] in his PhD thesis [16].

Using our generalised version of Equation (1.1) we are able to prove our main
result: the six-term exact sequence in $K$-theory

${\displaystyle\bigoplus_{[v]\in G\setminus
E^{0}}K_{0}(C^{*}_{r}(G_{v}))}$${\displaystyle\bigoplus_{[v]\in G\setminus
E^{0}}K_{0}(C^{*}_{r}(G_{v}))}$${K_{0}(C(\partial
E)\rtimes_{r}G)}$${K_{1}(C(\partial
E)\rtimes_{r}G)}$${\displaystyle\bigoplus_{[v]\in G\setminus
E^{0}}K_{1}(C^{*}_{r}(G_{v}))}$${\displaystyle\bigoplus_{[v]\in G\setminus
E^{0}}K_{1}(C^{*}_{r}(G_{v})),}$

where we give concrete formulae for the maps involved, which are induced from
homomorphisms on the level of the underlying $C^{*}$-algebras.

We apply this six-term exact sequence to produce specific $K$-theory formulae
for two classes of group actions on multitrees. We show that when $G$ acts
freely on $E$, then

$K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)\cong\operatorname{coker}(1-A^{T})\quad\text{and}\quad
K_{1}(C_{0}(\partial E)\rtimes_{\tau_{\partial},r}G)\cong\ker(1-A^{T}),$

where $A$ is the adjacency matrix of the quotient graph $G\backslash E$. In
the case where the action $G\curvearrowright E$ produces infinite cyclic
vertex stabiliser groups, and that $G\curvearrowright(E^{0}\cup\partial E)$ is
amenable, we show that

$K_{0}(C_{0}(\partial
E)\rtimes_{r}G)\cong\operatorname{coker}(1-A_{0})\oplus\ker(1-A_{1})$

and

$K_{1}(C_{0}(\partial
E)\rtimes_{r}G)\cong\operatorname{coker}(1-A_{1})\oplus\ker(1-A_{0}),$

where $A_{0}$ and $A_{1}$ are integer-valued matrices depending on the
transversal data obtained from the inclusions of edge stabilisers into vertex
stabilisers. This can be applied for example to generalised Baumslag–Solitar
groups acting on the boundary of directed trees.

Our work was initially motivated by the problem of calculating the $K$-theory
of the crossed products arising from group actions on undirected trees, and we
are able to use our results to answer this question. The link between
multitrees and undirected trees is given by the dual graph construction: if
$T$ is an undirected tree, then the dual graph $\widehat{T}$ is a multitree,
and there is a natural isomorphism of crossed products $C_{0}(\partial
T)\rtimes_{r}G\cong C_{0}(\partial\widehat{T})\rtimes_{r}G$. In the setting
where all vertex stabilisers are amenable, this means we have a six-term exact
sequence for the $K$-theory of the graph-of-groups $C^{*}$-algebras of [8],
and it connects our results to the $K$-theoretic results of Mundey and Rennie
for graph-of-groups $C^{*}$-algebras given in [17]. We do note that the two
approaches are different; while in [17] the maps in their six-term exact
sequence are described in terms of the $KK$-classes of
$C^{*}$-correspondences, we are able to give the more direct approach of
having induced homomorphisms as the maps in our sequence. It is clear that in
the setting of a free action, or when the vertex stabilisers are infinite
cyclic, our results coincide. In more generality it is not so clear, and it
would be an interesting problem to clarify the precise relationship between
the two approaches.

We start with preliminaries in Section 2, where we give the necessary
definitions and results for group actions on graphs. We also recall the
definition of the boundary of a locally finite, nonsingular tree, and its
topology. In Section 3 we look at the results of [11] on the $K$-theory of
crossed products induced by group actions on totally disconnected spaces, and
we generalise some of their results in the setting of discrete group actions.
The main tool to come out of work in this section is the generalised version
of Equation (1.1). Section 4 is devoted to the main results of the paper, on
the $K$-theory of the crossed product $C_{0}(\partial E)\rtimes_{r}G$ coming
from a discrete group $G$ acting on a row-finite, finitely aligned multitree
$E$ with no sources. We consider two classes of examples: when the action of
$G$ on $E$ is free, and hence the vertex stabilisers are trivial; and when all
vertex stabilisers are infinite cyclic. In both settings we give an explicit
formula for the $K$-theory of $C_{0}(\partial E)\rtimes_{r}G$ in terms of the
kernel and cokernel of a matrix (or matrices) associated to the underlying
data. In Section 5 we apply our $K$-theory results for multitrees to the dual
of an undirected tree $T$ to get an analogous six-term exact sequence in
$K$-theory for the crossed product from an action on the boundary of an
undirected tree.

We devote Section 6, our final section, to studying dynamical properties of
actions on multitrees. It is well known that dynamical properties of a
discrete group $G$ acting on a locally compact Hausdorff space $X$ induce key
structural properties of the corresponding reduced crossed product
$C_{0}(X)\rtimes_{r}G$. For example, $C_{0}(X)\rtimes_{r}G$ is simple if the
action is minimal and topologically free; a simple $C_{0}(X)\rtimes_{r}G$ is
purely infinite if the action is locally contractive; and the full and reduced
crossed products coincide if the action is amenable. For a countable discrete
group $G$ acting on a multitree $E$, we:

  * •

characterise minimality of $G\curvearrowright\partial E$ in terms of the
quotient directed graph $\Gamma:=G\backslash E$;

  * •

provide a sufficient condition on $\Gamma$ and the stabiliser subgroups that
ensures that $G\curvearrowright\partial E$ is locally contractive;

  * •

show that if $G\curvearrowright E^{0}$ is free, then
$G\curvearrowright\partial E$ is topologically free if and only if $\Gamma$ is
aperiodic;

  * •

characterise topological freeness of $G\curvearrowright\partial E$ in terms of
$\Gamma$ and the stabiliser subgroups in the case where these stabiliser
subgroups are infinite cyclic; and

  * •

in the case where $E$ is a directed tree $T_{+}$, show that if each stabiliser
subgroup is amenable, then the action $G\curvearrowright\partial T_{+}$ is
amenable.

Acknowledgements. The first author was supported by the Australian Research
Council grant DP200100155. The last author was supported by an Australian
Government Research Training Program Stipend Scholarship.

##  2\. Preliminaries on group actions on graphs

###  2.1. Graphs and trees

An undirected graph $\Gamma=(\Gamma^{0},\Gamma^{1},r,s)$ (called simply a
graph in [8, 17, 23]) consists of countable sets of vertices $\Gamma^{0}$ and
edges $\Gamma^{1}$ together with range and source maps
$r,s\colon\Gamma^{1}\to\Gamma^{0}$, as well as an involutive “edge reversal”
map $e\mapsto\overline{e}$ on $\Gamma^{1}$. We insist that $e\neq\overline{e}$
and $r(e)=s(\overline{e})$. We say that an undirected graph $\Gamma$ is
locally finite if $\lvert r^{-1}(v)\rvert<\infty$ for all $v\in\Gamma^{0}$,
and nonsingular if $\lvert r^{-1}(v)\rvert>1$ for all $v\in\Gamma^{0}$. All
undirected graphs in this paper will be locally finite and nonsingular.

A path (of length $n$) in $\Gamma$ is either a vertex $v\in\Gamma^{0}$ (when
$n=0$) or, if $n>0$, a sequence of edges $e_{1}e_{2}\dots e_{n}$ with
$s(e_{i})=r(e_{i+1})$ for $1\leq i\leq n-1$ (we note that we are using the
right-to-left “Australian” convention for notating paths, to be consistent
with [8] and [17]). A path $e_{1}e_{2}\dots e_{n}$ is reduced if either $n=0$,
or $e_{i+1}\neq\overline{e_{i}}$ for $1\leq i\leq n-1$ (that is, there is no
back-tracking in the path). We write $\Gamma^{n}$ (respectively,
$\Gamma^{n}_{r}$) for the collection of all paths (respectively, all reduced
paths) of length $n$ in $\Gamma$, and we write $\Gamma^{*}_{r}$ for the
collection of all reduced paths of any length in $\Gamma$. For any vertex
$v\in\Gamma^{0}$, we write $v\Gamma^{*}_{r}$ (respectively $v\Gamma^{n}_{r}$)
for the collection of reduced paths in $\Gamma$ (respectively, the collection
of reduced paths of length $n$) with range $v$, and we write $\Gamma^{*}_{r}v$
(respectively $\Gamma^{n}_{r}v$) for the collection of reduced paths in
$\Gamma$ (respectively, the collection of reduced paths of length $n$) with
source $v$.

We call a graph $\Gamma$ connected if for any two vertices $v,w\in\Gamma^{0}$
there is a path in $\Gamma$ between them. We call $\Gamma$ a tree if for any
two vertices $v,w\in\Gamma^{0}$, there is a unique reduced path from $v$ to
$w$.

We will also be considering directed graphs. A directed graph
$E=(E^{0},E^{1},r,s)$ consists of countable sets of vertices $E^{0}$ and edges
$E^{1}$ together with range and source maps $r,s\colon E^{1}\to E^{0}$, but
without an edge reversal map. Equivalently, we can think of a directed graph
$E$ as an undirected graph $\Gamma$ (called the underlying undirected graph of
$E$) equipped with an orientation, which is a subset $\Gamma_{+}^{1}$ of
$\Gamma^{1}$ which contains exactly one element of $\\{e,\overline{e}\\}$ for
each $e\in\Gamma^{1}$; this orientation specifies a direction for each edge in
the graph. A directed tree is a directed graph whose underlying undirected
graph is a tree.

Paths in a directed graph are defined and notated in the same way as for
undirected graphs. If we form a directed graph from an orientation on an
undirected graph, then paths can only consist of edges in the orientation.
Note however that the concept of a reduced path is redundant for directed
graphs, since the reversal of an edge in an orientation cannot be an element
of that orientation. We will therefore omit the subscript ‘$r$’ in the
$\Gamma^{*}_{r}$ notation when $\Gamma$ is directed.

We say that a directed graph $E$ is row-finite if $\lvert
r^{-1}(v)\rvert<\infty$ for all $v\in E^{0}$, and we say that a vertex $v\in
E^{0}$ is a source if $\lvert r^{-1}(v)\rvert=0$. All directed graphs in this
paper will be row-finite and have no sources.

###  2.2. Group actions on graphs

A group $G$ is said to act (on the left) on a graph $\Gamma$ (undirected or
directed) if it acts on the sets $\Gamma^{0}$ and $\Gamma^{1}$ in such a way
that $r(g\cdot e)=g\cdot r(e)$, $s(g\cdot e)=g\cdot s(e)$ and (for undirected
$\Gamma$) $\overline{g\cdot e}=g\cdot\overline{e}$ for all $e\in\Gamma^{1}$
and $g\in G$. For each $v\in\Gamma^{0}$, we write $G_{v}\leq G$ for the
stabiliser subgroup of $G$ at $v$, and similarly for each $e\in\Gamma^{1}$ we
write $G_{e}\leq G$ for the stabiliser subgroup of $G$ at $e$. These
stabiliser subgroups have the following properties:

  1. (1)

If $v,w\in\Gamma^{0}$ are in the same orbit under the action of $G$, then the
subgroups $G_{v}$ and $G_{w}$ are conjugate. Indeed, if $g\in G$ satisfies
$g\cdot v=w$, then $G_{w}=gG_{v}g^{-1}$. Similarly, if $e,f\in\Gamma^{1}$ are
in the same orbit under the action of $G$, then the subgroups $G_{e}$ and
$G_{f}$ are conjugate.

  2. (2)

For any $e\in\Gamma^{1}$, the group $G_{e}$ is a subgroup of both $G_{r(e)}$
and $G_{s(e)}$. This is because any automorphism of $\Gamma$ fixing $e$ must
also fix both the vertices $r(e)$ and $s(e)$.

  3. (3)

By the orbit-stabiliser theorem, for any $e\in\Gamma^{1}$, the orbit
$G_{r(e)}\cdot e$ is in one-to-one correspondence with the set of cosets
$G_{r(e)}/G_{e}$. In particular, if $\Gamma$ is locally finite or row-finite,
then $G_{e}$ is a finite-index subgroup of $G_{r(e)}$.

If a group $G$ acts on the directed graph $E$, then we can also consider the
quotient directed graph $G\backslash E$ associated to this action, which is
defined as follows. The vertices $(G\backslash E)^{0}$ are the vertex orbits
$\\{G\cdot v:v\in E^{0}\\}$; the edges $(G\backslash E)^{1}$ are the edge
orbits $\\{G\cdot e:e\in E^{1}\\}$; and the range and source maps are given by
$r(G\cdot e)=G\cdot r(e)$ and $s(G\cdot e)=G\cdot s(e)$, respectively, for
$e\in E^{1}$.

###  2.3. Boundaries

We recall the definition of the boundary of a locally finite, nonsingular
tree.

An infinite path in a graph $\Gamma$ (undirected or directed) is an infinite
sequence of edges $e_{1}e_{2}\dots$ such that $s(e_{i})=r(e_{i+1})$ for all
$i\geq 1$. Like for finite paths, we call an infinite path $e_{1}e_{2}\dots$
reduced if $e_{i+1}\neq\overline{e_{i}}$ for $i\geq 1$. We write
$\Gamma^{\infty}$ (respectively, $\Gamma^{\infty}_{r}$) for the collection of
all infinite paths (respectively, all reduced infinite paths) in $\Gamma$. For
any $v\in\Gamma^{0}$, we write $v\Gamma^{\infty}_{r}$ for the collection of
all reduced infinite paths with range $v$. For any infinite path
$\lambda=e_{1}e_{2}\dots$ in $\Gamma$ and any integer $n\geq 1$, we write
$\sigma^{n}(\lambda)$ for the infinite path $e_{n+1}e_{n+2}\dots$, and
$\lambda_{n}$ for the finite path $e_{1}\ldots e_{n}$.

If $T$ is a locally finite, nonsingular tree, then we can define its boundary
as follows. Define an equivalence relation on $T^{\infty}_{r}$ by
$\lambda\sim\mu$ if and only if $\sigma^{m}(\lambda)=\sigma^{n}(\mu)$ for some
$m,n\in\mathbb{N}$ (in this case we say that $\lambda$ and $\mu$ are shift-
tail equivalent). The boundary of $T$ is the set of equivalence classes of
reduced infinite paths in $T$, and is denoted $\partial T$. For $\lambda\in
T^{\infty}_{r}$ we write $[\lambda]$ for its shift-tail equivalence class in
$\partial T$.

We can put a topology on $\partial T$ in the following way. For a path
$\alpha\in T^{*}_{r}$ of length $n$, we say that an infinite path $\lambda\in
T^{\infty}_{r}$ extends $\alpha$ if $\lambda_{n}=\alpha$, and we define the
cylinder set $Z_{\partial}(\alpha)$ to be the subset $\\{[\lambda]:\lambda\in
T^{\infty}_{r}\ \text{extends}\ \alpha\\}$ of $\partial T$. The collection
$\\{Z_{\partial}(e):e\in T^{1}\\}$ is a base for a topology on $\partial T$,
turning it into a totally disconnected compact Hausdorff space.

If $T$ is directed, then the boundary is defined as above, and now
$\\{Z_{\partial}(v):v\in T^{0}\\}$ is a base for a totally disconnected
locally compact Hausdorff topology.

An action of a group $G$ on a tree $T$ naturally induces an action of the same
group on the boundary of tree, by $g\cdot[\lambda]=[g\cdot\lambda]$ for all
$\lambda\in T^{\infty}_{r}$. In this paper, we are interested in understanding
actions arising in this way.

##  3\. $K$-Theory of crossed products by actions on totally disconnected
spaces

In this section, we generalise the $K$-theory results of [11, Section 3] so
that we can apply them to the setting of group actions on multitrees in
Section 4.

Let $G$ be a second countable locally compact group acting continuously on a
second countable totally disconnected locally compact space $\Omega$, with
$\mathcal{V}=\\{V_{i}:i\in I\\}$ a $G$-invariant family of compact open sets
in $\Omega$. The $G$-invariance of $\mathcal{V}$ means that $G$ acts on $I$
with $g\cdot V_{i}=V_{g\cdot i}$ for each $g\in G,i\in I$. Denote by $\tau$
the induced action of $G$ on $C_{0}(\Omega)$, and suppose that $\alpha$ is an
action of $G$ on a separable $C^{*}$-algebra $A$. Under some additional
assumptions, in [11, Corollary 3.14] Cuntz, Echterhoff and Li prove that there
is an isomorphism

$K_{*}((A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G)\cong\bigoplus_{[i]\in
G\backslash I}K_{*}(A\rtimes_{\alpha,r}G_{i}),$

where $G_{i}\leq G$ is the stabiliser subgroup at $i\in I$. Our main result in
this section is Theorem 3.9, which establishes this isomorphism in a more
general setting than the one considered in [11].

We start with a discussion of this more general setting in Section 3.1, and in
Section 3.2 we prove a technical result which we then use in Section 3.3 to
generalise the $K$-theoretic arguments in [11].

###  3.1. Finite alignment

Let $X$ be a set, and $\mathcal{V}$ a subset of the power set
$\mathcal{P}(X)$. Following [11, Definition 2.4] we say $\mathcal{V}$ is
independent if for all $V,V_{1}\dots,V_{k}\in\mathcal{V}$ we have
$V=\cup_{i=1}^{k}V_{k}\implies V=V_{i}$ for some $1\leq i\leq k$. We say
$\mathcal{V}$ is finitely aligned if for all $V_{i},V_{j}\in\mathcal{V}$, the
intersection $V_{i}\cap V_{j}$ can be written as a (possibly empty) finite
disjoint union of other sets in $\mathcal{V}$.

In this subsection, we study the setting of a group $G$ acting on a totally
disconnected space $\Omega$ which admits a $G$-invariant, independent,
finitely aligned generating family of compact open subsets. Our main result
here is Proposition 3.2, which is a technical result that we will use to prove
Theorem 3.9.

######  Definition 3.1.

Let $H$ be any subgroup of $G$.

  1. (1)

For any $J\subseteq I$ define $\mathcal{V}_{J}:=\\{V_{i}:i\in J\\}$.

  2. (2)

Define $\mathcal{F}_{H}:=\\{F\subseteq I:F\text{ is finite and
$H$-invariant}\\}$.

  3. (3)

Define $\mathcal{J}_{H}:=\\{J\in\mathcal{F}_{H}:\mathcal{V}_{J}\text{ is
finitely aligned}\\}$.

######  Proposition 3.2.

Let $G$ be a second countable locally compact group acting continuously on a
second countable totally disconnected locally compact space $\Omega$. Let $I$
be a set, and let $\mathcal{V}:=\\{V_{i}:i\in I\\}$ be a $G$-invariant
collection of compact open subsets of $\Omega$ such that:

  1. (1)

$\mathcal{V}$ generates the compact open sets of $\Omega$ (via finite unions,
finite intersections, and differences);

  2. (2)

$\mathcal{V}$ is independent; and

  3. (3)

$\mathcal{V}$ is finitely aligned.

The set $\mathcal{J}_{H}$ is a cofinal subset of
$(\mathcal{F}_{H},\subseteq)$.

######  Remark 3.3.

Let $H$ be any compact subgroup of $G$. These assumptions on $\mathcal{V}$ in
the statement of Proposition 3.2 generalise the notion of a regular basis in
[11, Definition 2.9], where instead of finite alignment (3), the authors
assume that $\mathcal{V}$ is closed under intersections (up to the empty set).

To prove Proposition 3.2 we need the following lemma.

######  Lemma 3.4.

Let $F\subseteq I$ be finite. Then $\bigcap_{i\in F}V_{i}$ decomposes uniquely
as a finite disjoint union of sets in $\mathcal{V}$.

###### Proof.

A standard inductive argument shows that $\mathcal{V}$ finitely aligned means
that the intersection of finitely many elements of $\mathcal{V}$ decomposes as
a (possibly empty) finite disjoint union of sets in $\mathcal{V}$. So we just
need to prove uniqueness. Suppose that

(3.1) $\bigcap_{i\in
F}V_{i}=\bigsqcup_{i=1}^{m}W_{i}=\bigsqcup_{j=1}^{n}W^{\prime}_{j},$

where $m,n\geq 1$ and each $W_{i},W^{\prime}_{j}\in\mathcal{V}$. We want to
prove that

$\\{W_{1},\dots,W_{m}\\}=\\{W^{\prime}_{1},\dots,W^{\prime}_{n}\\}.$

Fix $i\in\\{1,\dots,m\\}$. By Equation (3.1), we have that

$W_{i}=\bigsqcup_{j=1}^{n}(W_{i}\cap W^{\prime}_{j}),$

where each $W_{i}\cap W^{\prime}_{j}$ is a disjoint union of sets in
$\mathcal{V}$. Then the independence of $\mathcal{V}$ implies that
$W_{i}\subseteq W_{i}\cap W^{\prime}_{j}$ for some $j$, which in turn implies
that $W_{i}\subseteq W^{\prime}_{j}$. By symmetry, there is some
$k\in\\{1,\dots,m\\}$ such that $W^{\prime}_{j}\subseteq W_{k}$, and so
$W_{i}\subseteq W_{k}$. But then disjointness of the sets $W_{1},\dots,W_{m}$
implies that we must have $i=k$, and so $W_{i}=W^{\prime}_{j}$. This shows
that
$\\{W_{1},\dots,W_{m}\\}\subseteq\\{W^{\prime}_{1},\dots,W^{\prime}_{n}\\}$,
and a symmetric argument gives equality. ∎

We are now able to prove Proposition 3.2.

###### Proof of Proposition 3.2.

Fix $F\in\mathcal{F}_{H}$. For each $Y\subseteq F$ we know from Lemma 3.4 that
there is a unique (finite) $J_{Y}\subseteq I$ with $\bigcap_{i\in
Y}V_{i}=\bigsqcup_{j\in J_{Y}}V_{j}$. We claim that $J:=\bigcup_{Y\subseteq
F}J_{Y}$ satisfies $F\subseteq J$ and $J\in\mathcal{J}_{H}$.

To see that $F\subseteq J$, let $i\in F$ and $Y=\\{i\\}$. Then
$J_{Y}=\\{i\\}$, and so $i\in J$. We now show that $J\in\mathcal{J}_{H}$. We
first claim that $J$ is $H$-invariant. Fix $j\in J$ and $h\in H$. Let
$Y\subseteq F$ with $j\in J_{Y}$, and consider $\bigcap_{i\in h\cdot Y}V_{i}$.
Since $F$ is $H$-invariant, we know that $h\cdot Y\subseteq F$, and so

$\bigcap_{i\in h\cdot Y}V_{i}=\bigsqcup_{k\in J_{h\cdot Y}}V_{k}.$

We also have

$\bigcap_{i\in h\cdot Y}V_{i}=\bigcap_{i\in Y}V_{h\cdot i}=\bigcap_{i\in
Y}h\cdot V_{i}=h\cdot\left(\bigcap_{i\in
Y}V_{i}\right)=h\cdot\left(\bigsqcup_{k\in J_{Y}}V_{k}\right)=\bigsqcup_{k\in
J_{Y}}h\cdot V_{k}.$

This gives two decompositions of $\bigcap_{i\in h\cdot Y}V_{i}$ as disjoint
unions of sets in $\mathcal{V}$; since by Lemma 3.4 such a decomposition is
unique, the set $h\cdot V_{j}$ appearing in the second decomposition must be
equal to $V_{k}$ for some $k\in J_{h\cdot Y}$. Hence $h\cdot j=k\in J$, and so
$J$ is $H$-invariant. Since $J$ is finite (being a finite union of finite
sets), we have $J\in\mathcal{F}_{H}$.

To prove that $\mathcal{V}_{J}$ is finitely aligned, we need to show that for
any $i,j\in J$, there is $J^{\prime}\subseteq J$ such that $V_{i}\cap
V_{j}=\bigsqcup_{k\in J^{\prime}}V_{k}$.

Let $Y,Y^{\prime}\subseteq F$ with $i\in J_{Y}$ and $j\in J_{Y^{\prime}}$, and
consider $\bigcap_{k\in Y\cup Y^{\prime}}V_{k}$. We have

$\bigcap_{k\in Y\cup Y^{\prime}}V_{k}=\bigsqcup_{l\in J_{Y\cup
Y^{\prime}}}V_{l}.$

We also have

$\bigcap_{k\in Y\cup Y^{\prime}}V_{k}=\left(\bigcap_{m\in
Y}V_{m}\right)\cap\left(\bigcap_{m^{\prime}\in
Y^{\prime}}V_{m^{\prime}}\right)=\left(\bigsqcup_{n\in
J_{Y}}V_{n}\right)\cap\left(\bigsqcup_{n^{\prime}\in
Y^{\prime}}V_{n^{\prime}}\right)=\bigsqcup_{n\in J_{Y},n^{\prime}\in
J_{Y^{\prime}}}(V_{n}\cap V_{n^{\prime}}).$

For each $n\in J_{Y}$ and $n^{\prime}\in J_{Y^{\prime}}$, finite alignment
ensures that there is a set $Y_{n,n^{\prime}}\subseteq I$ with $V_{n}\cap
V_{n^{\prime}}=\bigsqcup_{p\in Y_{n,n^{\prime}}}V_{p}$, so

$\bigcap_{k\in Y\cup Y^{\prime}}V_{k}=\bigsqcup_{\begin{subarray}{c}n\in
J_{Y},n^{\prime}\in J_{Y^{\prime}}\\\ p\in
Y_{n,n^{\prime}}\end{subarray}}V_{p}.$

So we have two decompositions of $\bigcap_{k\in Y\cup Y^{\prime}}V_{k}$ as
disjoint unions of sets in $\mathcal{V}$, and since by Lemma 3.4 such a
decomposition is unique, we get that

$V_{i}\cap V_{j}=\bigsqcup_{p\in Y_{i,j}}V_{p}=\bigsqcup_{l\in
J^{\prime}\subseteq J_{Y\cup Y^{\prime}}}V_{l}$

for some subset $J^{\prime}\subseteq J_{Y\cup Y^{\prime}}\subseteq J$. Hence
$\mathcal{V}_{J}$ is finitely aligned. ∎

###  3.2. Results on projections

We know from [11, Lemmas 2.2 and 2.3] that studying sets of compact open
subsets of a totally disconnected space which generate all compact open
subsets is equivalent to studying sets of projections generating a commutative
$C^{*}$-algebra. This subsection is dedicated to proving Proposition 3.6,
which is a generalisation of [11, Lemma 3.8], and uses the language of
projections and $C^{*}$-algebras.

######  Notation 3.5.

For $\\{e_{1},\dots,e_{n}\\}$ a set of projections, for each $1\leq i\leq n$
we define

$e^{\prime}_{i}:=e_{i}-\bigvee_{e_{j}<e_{i}}e_{j}.$

######  Proposition 3.6.

Let $\\{e_{1},\dots,e_{n}\\}$ be a finite set of commuting projections
generating a $C^{*}$-algebra $D$.

  1. (1)

The set $\\{e^{\prime}_{1},\dots,e^{\prime}_{n}\\}$ is a family of non-zero,
pairwise orthogonal projections linearly spanning $D$ if and only if
$\\{e_{1},\dots,e_{n}\\}$ is independent and $e_{i}e_{j}=\bigvee_{e_{k}\leq
e_{i}e_{j}}e_{k}$ for any $i,j\in\\{1,\dots,n\\}$.

  2. (2)

In this case, the transition matrix $\Gamma=(\gamma_{ij})$ determined by the
equations $e_{j}=\sum_{i=1}^{n}\gamma_{ij}e_{i}^{\prime}$ is a unipotent,
$\\{0,1\\}$-matrix, and hence invertible over $\mathbb{Z}$.

To prove Proposition 3.6, we need the following lemma.

######  Lemma 3.7.

Let $\\{e_{1},\dots,e_{n}\\}$ be a finite set of commuting projections
generating a $C^{*}$-algebra $D$. Suppose that
$\\{e^{\prime}_{1},\dots,e^{\prime}_{n}\\}$ is a family of non-zero, pairwise
orthogonal projections linearly spanning $D$. Then $e^{\prime}_{a}\leq
e_{i}\iff e_{a}\leq e_{i}$, for any $a,i\in\\{1,\dots,n\\}$.

###### Proof.

The implication $e_{a}\leq e_{i}\implies e^{\prime}_{a}\leq e_{i}$ is
immediate. For the reverse implication, note that we can write each $e_{i}$ as
$e_{i}=\sum_{a=1}^{n}\gamma_{ai}e_{a}^{\prime}$, where for each $a=1,\dots,n$,
we have $\gamma_{ai}=1$ if $e^{\prime}_{a}\leq e_{i}$ and $\gamma_{ai}=0$
otherwise. Pairwise orthogonality of the $e^{\prime}_{i}$ means that for each
pair of distinct $i,j\in\\{1,\dots,n\\}$, we have

$0=e^{\prime}_{i}e^{\prime}_{j}=\left(e_{i}-\bigvee_{e_{k}<e_{i}}e_{k}\right)\left(e_{j}-\bigvee_{e_{l}<e_{j}}e_{l}\right)=e_{i}e_{j}-\bigvee_{e_{l}<e_{j}}e_{i}e_{l}-\bigvee_{e_{k}<e_{i}}e_{j}e_{k}+\bigvee_{\begin{subarray}{c}e_{k}<e_{i}\\\
e_{l}<e_{j}\end{subarray}}e_{k}e_{l},$

and so

$\displaystyle e_{i}e_{j}$
$\displaystyle=\bigvee_{e_{l}<e_{j}}e_{i}e_{l}+\bigvee_{e_{k}<e_{i}}e_{j}e_{k}-\bigvee_{\begin{subarray}{c}e_{k}<e_{i}\\\
e_{l}<e_{j}\end{subarray}}e_{k}e_{l}$
$\displaystyle=\bigvee_{e_{l}<e_{j}}\left(\sum_{a=1}^{n}\gamma_{ai}\gamma_{al}e^{\prime}_{a}\right)+\bigvee_{e_{k}<e_{i}}\left(\sum_{a=1}^{n}\gamma_{aj}\gamma_{ak}e^{\prime}_{a}\right)-\bigvee_{\begin{subarray}{c}e_{k}<e_{i}\\\
e_{l}<e_{j}\end{subarray}}\left(\sum_{a=1}^{n}\gamma_{ak}\gamma_{al}e^{\prime}_{a}\right).$

If we define $\mu_{ai}:=\max\\{\gamma_{ak}:e_{k}<e_{i}\\}$ for all
$a,i\in\\{1,\dots,n\\}$, then

(3.2)
$e_{i}e_{j}=\sum_{a=1}^{n}\gamma_{ai}\mu_{aj}e^{\prime}_{a}+\sum_{a=1}^{n}\gamma_{aj}\mu_{ai}e^{\prime}_{a}-\sum_{a=1}^{n}\mu_{ai}\mu_{aj}e^{\prime}_{a}=\sum_{a=1}^{n}(\gamma_{ai}\mu_{aj}+\gamma_{aj}\mu_{ai}-\mu_{ai}\mu_{aj})e^{\prime}_{a}.$

Now suppose that $e^{\prime}_{a}\leq e_{i}$. If $a=i$ then $e_{a}\leq e_{i}$
trivially, so suppose that $a\neq i$. Multiplying both sides of the inequality
$e^{\prime}_{a}\leq e_{i}$ by $e_{a}$ gives that
$e^{\prime}_{a}=e_{a}e^{\prime}_{a}\leq e_{a}e_{i}$. From Equation (3.2), this
implies that

$\gamma_{aa}\mu_{ai}+\gamma_{ai}\mu_{aa}-\mu_{aa}\mu_{ai}=1.$

Now, $\gamma_{aa}=1$ by definition, since $e^{\prime}_{a}\leq e_{a}$. We also
claim that $\mu_{aa}=0$. Indeed, the definition of $\mu_{aa}$ is

$\mu_{aa}=\max\\{\gamma_{ak}:e_{k}<e_{a}\\}.$

But if $e_{k}<e_{a}$, then by the definition of $e^{\prime}_{a}$ we have that
$e_{k}e^{\prime}_{a}=0$, and so $\gamma_{ak}=0$. So $\mu_{aa}=0$. This implies
that

$(1)\mu_{ai}+\gamma_{ai}(0)-(0)\mu_{ai}=\mu_{ai}=1.$

By definition of $\mu_{ai}$, this means that there is some $j$ such that

$e^{\prime}_{a}\leq e_{j}<e_{i}.$

If $j=a$, then $e^{\prime}_{a}\leq e_{a}<e_{i}$, and we are done. So suppose
$j\neq a$. Then repeating the above argument with $j$ instead of $i$ yields
some $k$ such that

$e^{\prime}_{a}\leq e_{k}<e_{j}<e_{i}.$

Once again, if $k=a$, then $e^{\prime}_{a}\leq e_{a}<\cdots<e_{i}$, and we are
done. Otherwise, we can repeatedly apply the above argument, and at each step
we have indices $j,k,\dots,b$ such that

$e^{\prime}_{a}\leq e_{b}<\cdots<e_{k}<e_{j}<e_{i}.$

Now since there are a finite number of projections, this process must
eventually terminate (otherwise we would get an infinite, strictly decreasing
chain of projections, which is impossible). So at some point the chain will
look like

$e^{\prime}_{a}\leq e_{a}<\cdots<e_{j}<e_{i},$

which shows that $e_{a}\leq e_{i}$ as desired. ∎

###### Proof of Proposition 3.6.

For (1), we first assume that $\\{e_{1},\dots,e_{n}\\}$ is independent and
$e_{i}e_{j}=\bigvee_{e_{k}\leq e_{i}e_{j}}e_{k}$ for any
$i,j\in\\{1,\dots,n\\}$. The independence of $\\{e_{1},\dots,e_{n}\\}$ is
equivalent to each $e^{\prime}_{i}$ being non-zero, essentially by definition.
Note also that for each $i\in\\{1,\dots,n\\}$, we have $e^{\prime}_{i}\leq
e_{i}$ (again, by definition). Fix distinct $i,j\in\\{1,\dots,n\\}$.
Independence implies that $e_{i}\neq e_{j}$. If $e_{j}<e_{i}$, then by
definition of $e^{\prime}_{i}$, we get that $e^{\prime}_{i}e_{j}=0$, which
implies that
$e^{\prime}_{i}e^{\prime}_{j}=e^{\prime}_{i}e_{j}e^{\prime}_{j}=0$. If $e_{j}$
is not less than $e_{i}$, then we must have $e_{i}e_{j}<e_{j}$. In this case,

$e_{i}e_{j}=\bigvee_{e_{k}\leq
e_{i}e_{j}}e_{k}\leq\bigvee_{e_{k}<e_{j}}e_{k},$

and multiplying through by $e^{\prime}_{j}$ gives that
$e_{i}e^{\prime}_{j}<0$, and hence $e_{i}e^{\prime}_{j}=0$. This in turns
implies that $e^{\prime}_{i}e^{\prime}_{j}=0$. Thus,
$e^{\prime}_{1},\dots,e^{\prime}_{n}$ are pairwise orthogonal. To see that
$e^{\prime}_{1},\dots,e^{\prime}_{n}$ linearly span $D$, it suffices to show
that $\dim D\leq n$. For this, note that $e_{i}e_{j}$ is either $e_{i}$ or
$e_{j}$, or is the join $\bigvee_{e_{k}<e_{i},e_{j}}e_{k}$, which is a linear
combination of products of the projections $e_{1},\dots,e_{n}$. If we keep
writing products as single projections or joins, then since there are only
finitely many $e_{1},\dots,e_{n}$, we eventually get $e_{i}e_{j}$ expressed as
a linear combination of these projections. An inductive argument gives that
any finite product of the $e_{i}$’s can be written as a linear combination of
the $e_{i}$’s, and it follows that $\dim D\leq n$.

Now suppose that $\\{e^{\prime}_{1},\dots,e^{\prime}_{n}\\}$ is a family of
non-zero, pairwise orthogonal projections linearly spanning $D$. Then each
$e_{i}e_{j}$, being a projection in $D$, can be expressed as a sum

$e_{i}e_{j}=\sum_{e^{\prime}_{a}\leq e_{i}e_{j}}e^{\prime}_{a},$

and so Lemma 3.7 implies that

$e_{i}e_{j}=\bigvee_{e^{\prime}_{a}\leq e_{i}e_{j}}e_{a},$

as required.

For (2), we just note that Lemma 3.7 gives $\gamma_{ij}=1\iff e_{i}\leq
e_{j}$, and then the arguments in the proof of [11, Lemma 3.8] give the
result. ∎

###  3.3. Results on $K$-theory

We now state and prove Theorem 3.9, which is a generalisation of [11,
Corollary 3.14 and Lemma 3.16].

######  Notation 3.8.

In the below statement, we denote by $\xi_{i}$ the standard basis element in
$C_{0}(I)$, and by $e_{i,j}$ the rank-one partial isometry in
$\mathcal{K}(\ell^{2}(I))$ which sends $\xi_{j}$ to $\xi_{i}$. We denote by
$u$ the canonical unitary representation of a group into any crossed product
induced from an action of the group.

######  Theorem 3.9.

Let $G$ be a second countable locally compact group acting continuously on a
second countable totally disconnected locally compact space $\Omega$, and
denote by $\tau$ the induced action of $G$ on $C_{0}(\Omega)$. Let $A$ be a
separable $C^{*}$-algebra, and $\alpha$ an action of $G$ on $A$. Suppose that
$\mathcal{V}=\\{V_{i}:i\in I\\}$ is a $G$-invariant collection of compact open
subsets of $\Omega$ that generates the compact open subsets of $\Omega$, and
is independent and finitely aligned. Suppose that $G$ satisfies the
Baum–Connes conjecture with coefficients in $A\otimes C_{0}(I)$ and $A\otimes
C_{0}(\Omega)$. Then there is a homomorphism

$\bigoplus_{[i]\in G\backslash
I}A\rtimes_{\alpha,r}G_{i}\to\mathcal{K}(\ell^{2}(I))\otimes\big{(}(A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G\big{)},$

which, when restricted to the $[i]$ summand, satisfies $au_{g}\mapsto
e_{i,i}\otimes((a\otimes 1_{V_{i}})u_{g})$ for all $a\in A,g\in G_{i}$, where
$G_{i}\leq G$ is the stabiliser subgroup at $i\in I$, under the action of $G$
on $I$ satisfying $g\cdot V_{i}=V_{g\cdot i}$ for all $g\in G,i\in I$.
Moreover, this homomorphism induces an isomorphism

$\bigoplus_{[i]\in G\backslash I}K_{*}(A\rtimes_{\alpha,r}G_{i})\cong
K_{*}((A\otimes C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G),$

which, when restricted to the $[i]$ summand, is the map in $K$-theory induced
from the inclusion $A\rtimes_{\alpha,r}G_{i}\to(A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G,\,au_{g}\mapsto(a\otimes
1_{V_{i}})u_{g}$. When $A=\mathbb{C}$, we get an isomorphism

$\bigoplus_{[i]\in G\backslash I}K_{*}(C^{*}_{r}(G_{i}))\cong
K_{*}(C_{0}(\Omega)\rtimes_{\tau,r}G).$

We denote by $\mu$ the action of $G$ on $C_{0}(I)$ induced from the action of
$G$ on $I$ described in Theorem 3.9. Since we are generalising [11, Corollary
3.14], we first discuss its proof. The proof is broken into two main parts:

  * (Step 1)

the homomorphism $\bigoplus_{[i]\in G\backslash
I}A\rtimes_{\alpha,r}G_{i}\to(A\otimes C_{0}(I))\rtimes_{\alpha\otimes\mu,r}G$
(called $\eta$ in [11, Remark 3.15]), which induces an isomorphism

$\bigoplus_{[i]\in G\backslash I}K_{*}(A\rtimes_{\alpha,r}G_{i})\cong
K_{*}((A\otimes C_{0}(I))\rtimes_{\alpha\otimes\mu,r}G);$

and

  * (Step 2)

the homomorphism $(A\otimes
C_{0}(I))\rtimes_{\alpha\otimes\mu,r}G\to\mathcal{K}(\ell^{2}(I))\otimes\big{(}(A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G\big{)}$ (which, up to isomorphism
of the codomain, is the map $\psi$ from [11, Remark 3.15]), which induces an
isomorphism

$K_{*}((A\otimes C_{0}(I))\rtimes_{\alpha\otimes\mu,r}G)\cong K_{*}((A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G).$

The result in (Step 1) applies in our setting, but we require some adjustments
to prove (Step 2) with a finitely-aligned $\mathcal{V}$.

The proof of (Step 2) in [11] proceeds as follows. The authors first use [11,
Proposition 3.3] to reduce the problem to actions involving compact subgroups
$H$ of $G$. Then in [11, Lemma 3.6] (see also [11, Remark 3.7] for the
translation between $H$-equivariant $K$-theory and the $K$-theory of the
corresponding crossed products by $H$), the problem is further reduced to
considering finite subsets of $\mathcal{V}$. In particular, for
$J\in\mathcal{F}_{H}$ (recall that $\mathcal{F}_{H}$ is the set of all finite
$H$-invariant subsets of $I$), and $C_{0}(\Omega)_{J}:=C^{*}(\\{1_{V_{j}}:j\in
J\\})\subseteq C_{0}(\Omega)$, if the maps $K_{*}(C_{0}(J)\rtimes H)\to
K_{*}(C_{0}(\Omega)_{J}\rtimes H)$ are isomorphisms for every $J$ in some
cofinal subset of the set $\mathcal{F}_{H}$ ordered by inclusion, then there
is an isomorphism

$K_{*}((A\otimes C_{0}(I))\rtimes_{\alpha\otimes\mu}H)\cong K_{*}((A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau}H).$

Finally, the key step in getting isomorphisms $K_{*}(C_{0}(J)\rtimes H)\cong
K_{*}(C_{0}(\Omega)_{J}\rtimes H)$ for every compact $H$ and every
$J\in\mathcal{F}_{H}$, is the following lemma:

######  Lemma 3.10 (Lemma 3.8 of [11]).

Let $D$ be a commutative $C^{*}$-algebra generated by a multiplicatively
closed (up to 0) and independent finite family of projections $\\{p_{i}:i\in
I\\}$. For each $i\in I$ let
$p^{\prime}_{i}:=p_{i}-\bigvee_{p_{j}<p_{i}}p_{j}$. Then
$\\{p^{\prime}_{i}\\}$ is a family of nonzero pairwise orthogonal projections
spanning $D$. The transition matrix $\Gamma=(\gamma_{ij})$ determined by the
equations $p_{j}=\sum_{i\in I}\gamma_{ij}p^{\prime}_{i}$ is unipotent and
therefore invertible over $\mathbb{Z}$.

In [11, Appendix A] the authors show how the matrix $\Gamma$ gives rise to a
class $x_{\Gamma}\in KK^{H}(C_{0}(J),C_{0}(\Omega)_{J})$, and prove that
$\Gamma$ being invertible implies that $x_{\Gamma}$ is invertible. Remark 3.9
of [11] then explains how $x_{\Gamma}$ being invertible ensures that the maps
$K_{*}(C_{0}(J)\rtimes H)\to K_{*}(C_{0}(\Omega)_{J}\rtimes H)$ are
isomorphisms; this argument relies on the entries of $\Gamma$ being elements
of $\\{0,1\\}$, which is shown in the proof of [11, Lemma 3.8].

We now state and prove our replacement of (Step 2) for our more general
setting. This, when combined with (Step 1), gives Theorem 3.9.

######  Theorem 3.11.

Let $G$ be a second countable locally compact group acting continuously on a
second countable totally disconnected locally compact space $\Omega$, and
denote by $\tau$ the induced action of $G$ on $C_{0}(\Omega)$. Let $A$ be a
separable $C^{*}$-algebra, and $\alpha$ an action of $G$ on $A$. Suppose that
$\mathcal{V}=\\{V_{i}:i\in I\\}$ is a $G$-invariant collection of compact open
subsets of $\Omega$ that generates the compact open subsets of $\Omega$, and
is independent and finitely aligned. Denote by $\mu$ the action of $G$ on
$C_{0}(I)$ induced from the action of $G$ on $I$ satisfying $g\cdot
V_{i}=V_{g\cdot i}$ for all $g\in G,i\in I$. Suppose that $G$ satisfies the
Baum–Connes conjecture with coefficients in $A\otimes C_{0}(I)$ and $A\otimes
C_{0}(\Omega)$. Then there is a homomorphism

$(A\otimes
C_{0}(I))\rtimes_{\alpha\otimes\mu,r}G\to\mathcal{K}(\ell^{2}(I))\otimes\big{(}(A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G\big{)}$

satisfying $(a\otimes\xi_{i})u_{g}\mapsto e_{i,g^{-1}\cdot i}\otimes((a\otimes
1_{V_{i}})u_{g})$ for all $a\in A,g\in G,i\in I$, which induces an isomorphism

$K_{*}((A\otimes C_{0}(I))\rtimes_{\alpha\otimes\mu,r}G)\cong K_{*}((A\otimes
C_{0}(\Omega))\rtimes_{\alpha\otimes\tau,r}G).$

###### Proof.

The above discussion of the proof of [11, Corollary 3.14] means we need to
provide: a cofinal subset of $\mathcal{F}_{H}$ such that for every $J$ in that
subset, the maps $K_{*}(C_{0}(J)\rtimes H)\to K_{*}(C_{0}(\Omega)_{J}\rtimes
H)$ are isomorphisms; and a suitable replacement for Lemma 3.10. For this we
take $\mathcal{J}_{H}$ from Proposition 3.2, and part (2) of Proposition 3.6.
∎

This completes the proof of Theorem 3.9.

##  4\. Group actions on multitrees

Our main application of Theorem 3.9 is in studying the $K$-theory of the
reduced crossed product of a group acting on the boundary of a certain type of
directed graph (a finitely aligned multitree). These graphs generalise
directed trees and also allow us to study group actions on the boundaries of
undirected trees, which will be discussed in Section 5.

We start with a discussion of multitrees in Section 4.1, and then in Section
4.2 we explain how Theorem 3.9 can be applied to this setting. Following this,
we give formulae for the $K$-theory of the crossed product in the special
cases where the group acts freely on the multitree (Section 4.3) and where the
vertex stabilisers of the action are infinite cyclic (Section 4.4).

###  4.1. Multitrees

In this subsection, we give a definition of multitrees, and prove some useful
results about their structure.

######  Definition 4.1.

A multitree is a directed graph $E=(E^{0},E^{1},r,s)$ such that for any two
vertices $v,w\in E^{0}$, there is at most one (directed) path from $w$ to $v$
(or equivalently, at most one directed path between $v$ and $w$). This
condition ensures that there is a partial order on the vertex set $E^{0}$
defined by $v\leq w$ if and only if there is a directed path from $w$ to $v$.
We call a multitree $E$ finitely aligned if for any pair of vertices $v,w\in
E^{0}$, the collection $v\vee w$ of minimal common upper bounds of $v$ and
$w$, under the partial order described above, is finite.

Multitrees have also been called strongly unambiguous graphs or mangroves in
[1].

######  Example 4.2 (Example of a multitree).

In this paper, we restrict our attention to multitrees $E$ which are row-
finite, finitely aligned and have no sources. We can define the boundary
$\partial E$ of a multitree $E$ in the same way as for directed trees (as
described in Section 2.3), and it will also be a totally disconnected locally
compact Hausdorff space. Given an action of a countable discrete group $G$ on
$E$, we wish to apply Theorem 3.9 to calculate the $K$-theory of the reduced
crossed product induced from the action of $G$ on the boundary $\partial E$.
However, this boundary will not have a suitable collection of compact open
subsets, so in order to apply the theorem, we will need to enlarge the space
with the vertices $E^{0}$.

######  Definition 4.3.

Suppose that $E$ is a row-finite, finitely aligned multitree without sources,
and consider the space $E^{0}\cup\partial E$. We define the cylinder set of a
vertex $v\in E^{0}$ to be $Z(v):=\\{w\in E^{0}:vE^{*}w\neq\emptyset\\}\cup
Z_{\partial}(v)$. The collection $\\{\\{v\\}:v\in E^{0}\\}\cup\\{Z(v):v\in
E^{0}\\}$ forms a basis for a topology of compact open sets turning
$E^{0}\cup\partial E$ into a totally disconnected locally compact Hausdorff
space.

The following proposition will be useful for Section 4.2.

######  Proposition 4.4.

Let $E$ be a row-finite, finitely aligned multitree with no sources. For any
two vertices $v,w\in E^{0}$, we have that

$Z(v)\cap Z(w)=\bigsqcup_{u\in v\vee w}Z(u).$

We will use the following lemma in the proof of Proposition 4.4.

######  Lemma 4.5.

Let $E$ be a row-finite, finitely aligned multitree with no sources, and let
$v,w\in E^{0}$. The following are equivalent:

  1. (i)

$v\leq w$;

  2. (ii)

$Z(w)\subseteq Z(v)$;

  3. (iii)

$w\in Z(v)$.

###### Proof.

We show that (i) $\implies$ (ii) $\implies$ (iii) $\implies$ (i).

First suppose (i), so $v\leq w$. This means that there is a path $\lambda\in
vE^{*}w$. Now take $x\in Z(w)$. We treat the two cases $x\in E^{0}$ and
$x\in\partial E$ separately. If $x\in E^{0}$, then by definition of $Z(w)$,
there is a path $\mu\in wE^{*}x$. So $\lambda\mu\in vE^{*}x$, which implies
that $x\in Z(v)$. If $x\in\partial E$, then $x=[\nu]$ for some $\nu\in
wE^{\infty}$. So $\lambda\nu\in vE^{\infty}$, and so $[\lambda\nu]=[\nu]=x$ is
again in $Z(v)$. This shows that $Z(w)\subseteq Z(v)$, so (ii) holds.

The implication (ii) $\implies$ (iii) follows immediately from the fact that
$w\in Z(w)$. Finally, suppose (iii), so $w\in Z(v)$. Since $w\in E^{0}$, this
means that there is a path $\lambda\in vE^{*}w$, which exactly means that
$v\leq w$, which is (i). ∎

###### Proof of Proposition 4.4.

Fix $v,w\in E^{0}$. First we show that

(4.1) $Z(v)\cap Z(w)=\bigcup_{u\in v\vee w}Z(u).$

For the backwards inclusion, fix a minimal common upper bound $u$ of $v$ and
$w$. Since both $v\leq u$ and $w\leq u$, Lemma 4.5 implies that $Z(u)\subseteq
Z(v)\cap Z(w)$. So the right-hand side of Equation (4.1) is also contained in
$Z(v)\cap Z(w)$.

For the forwards inclusion, take $x\in Z(v)\cap Z(w)$. We treat the two cases
$x\in E^{0}$ and $x\in\partial E$ separately. First suppose that $x\in E^{0}$.
Then by Lemma 4.5, we have that $v\leq x$ and $w\leq x$. So $x$ is a common
upper bound of $v$ and $w$, which means that there must be some minimal common
upper bound $u$ of $v$ and $w$ such that $u\leq x$. Then Lemma 4.5 gives that
$x\in Z(u)$. Now suppose that $x=[\lambda]\in\partial E$, for some $\lambda\in
E^{\infty}$. This means that there are infinite paths $\lambda_{v}\in
vE^{\infty}$ and $\lambda_{w}\in wE^{\infty}$ such that $\lambda$,
$\lambda_{v}$ and $\lambda_{w}$ are shift-tail equivalent to each other. So
there is an infinite path $\mu\in E^{\infty}$ and finite paths $\nu_{v}\in
vE^{*}r(\mu)$ and $\nu_{w}\in wE^{*}r(\mu)$ such that $\lambda_{v}=\nu_{v}\mu$
and $\lambda_{w}=\nu_{w}\mu$. Then the vertex $r(\mu)$ is a common upper bound
of $v$ and $w$, and by the same argument as in the first case, there is some
minimal common upper bound $u$ of $v$ and $w$ such that $u\leq r(\mu)$. Let
$\nu_{u}$ be the path from $r(\mu)$ to $u$. Then $\nu_{u}\mu\in uE^{\infty}$
is shift-tail equivalent to $\lambda$ and so $x=[\lambda]=[\nu_{u}\mu]$ lies
in $Z(u)$. This proves the forwards inclusion and hence we have shown Equation
(4.1).

It remains to prove that the union $\cup_{u}Z(u)$ is disjoint. Suppose for a
contradiction that there are two distinct minimal common upper bounds $u_{1}$
and $u_{2}$ of $v$ and $w$ such that $Z(u_{1})\cap Z(u_{2})\neq\emptyset$.
Equation (4.1), with $u_{1}$ and $u_{2}$ in place of $v$ and $w$ respectively,
implies that $u_{1}$ and $u_{2}$ have a (minimal) common upper bound
$u^{\prime}\in E^{0}$. So there are paths $\lambda_{1}\in
u_{1}E^{*}u^{\prime}$ and $\lambda_{2}\in u_{2}E^{*}u^{\prime}$. Also, since
$u_{1}$ and $u_{2}$ are both upper bounds of $v$, there are paths $\mu_{1}\in
vE^{*}u_{1}$ and $\mu_{2}\in vE^{*}u_{2}$. So then we have two paths
$\mu_{1}\lambda_{1}$ and $\mu_{2}\lambda_{2}$ which both go from $u^{\prime}$
to $v$. Moreover, the paths $\mu_{1}\lambda_{1}$ and $\mu_{2}\lambda_{2}$ must
be distinct, since otherwise $u_{1}$ and $u_{2}$ would lie on the same
directed path and would therefore be comparable, contradicting the fact that
$u_{1}$ and $u_{2}$ are distinct minimal common upper bounds of $u$. But since
there are two distinct paths from $u^{\prime}$ to $v$, this contradicts the
fact that $E$ is a multitree, and so the intersection $Z(u_{1})\cap Z(u_{2})$
must be empty. This shows that the union $\cup_{u}Z(u)$ is disjoint. ∎

###  4.2. $K$-theory

Suppose a group $G$ acts on a row-finite, finitely aligned multitree $E$ with
no sources. Our main result of this section is Theorem 4.8, which describes
the $K$-theory of the reduced crossed product of $G$ acting on the boundary of
$E$. Before we can state the theorem, we introduce some notation.

######  Notation 4.6.

In this subsection, we will always write $\Omega$ for the topological space
$E^{0}\cup\partial E$ from Definition 4.3. We will write $\tau$ for the
induced action of $G$ on $C_{0}(\Omega)$, and we will write $\tau_{0}$ and
$\tau_{\partial}$ for the restriction of $\tau$ to $C_{0}(E^{0})$ and
$C_{0}(\partial E)$ respectively. For each $v\in E^{0}$, we write $\xi_{v}\in
C_{0}(E^{0})$ for the indicator function of the singleton set $\\{v\\}$, and
we write $1_{Z(v)}\in C_{0}(\Omega)$ for the indicator function of the set
$Z(v)\subseteq\Omega$.

For each $e\in E^{1}$, we will fix some transversal $\Sigma_{e}$ for
$G_{r(e)}/G_{e}$ containing the identity element $1_{G_{r(e)}}$ (which is just
$1_{G}$). Note that $\Sigma_{e}$ has a natural left action of $G_{r(e)}$
defined by setting $g\cdot h$ to be the representative in $\Sigma_{e}$ of the
coset $ghG_{e}$, for $g\in G_{r(e)}$, $h\in\Sigma_{e}$. For
$h,k\in\Sigma_{e}$, we denote by $e_{h,k}\in\mathcal{K}(\ell^{2}(\Sigma_{e}))$
the rank-one partial isometry sending the standard basis element
$\xi_{k}\in\ell^{2}(\Sigma_{e})$ at $k$ to the standard basis element
$\xi_{h}$ at $h$. We denote by $u$ the canonical unitary representation of a
group into its reduced group $C^{*}$-algebra. Finally, we write $\theta_{e}$
for the map defined by

$\displaystyle\theta_{e}\colon C^{*}_{r}(G_{r(e)})$
$\displaystyle\to\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes C^{*}_{r}(G_{s(e)})$
$\displaystyle u_{g}$
$\displaystyle\mapsto\sum_{h\in\Sigma_{e}}e_{h,g^{-1}\cdot h}\otimes
u_{h^{-1}g(g^{-1}\cdot h)}.$

Note that for any $g\in G_{r(e)}$ and $h\in\Sigma_{e}$, the elements $g^{-1}h$
and $g^{-1}\cdot h$ are in the same coset in $G_{r(e)}/G_{e}$, so the element
$h^{-1}g(g^{-1}\cdot h)$ is in $G_{e}$ and therefore also in $G_{s(e)}$. Also,
recall that the row-finiteness of $E$ implies that $\Sigma_{e}$ is finite, so
the sum in the above formula for $\theta_{e}$ is well-defined.

We are primarily interested in the induced map $(\theta_{e})_{*}$ on
$K$-theory; the following lemma shows that this map is (essentially)
independent of the choice of $e\in E^{1}$ in a fixed edge orbit. Note that the
lemma also implies that $(\theta_{e})_{*}$ is independent of the choice of
transversal $\Sigma_{e}$, by taking $e^{\prime}=e$ and $g$ to be the identity
element of $G$.

######  Lemma 4.7.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources. Let $e,e^{\prime}\in E^{1}$ be edges in the
same orbit, and let $g\in G$ be an element such that $e^{\prime}=g\cdot e$.
Write $\operatorname{Ad}u_{g}\colon C^{*}_{r}(G_{r(e)})\to
C^{*}_{r}(G_{r(e^{\prime})})$ for the *-homomorphism defined by $u_{h}\mapsto
u_{ghg^{-1}}$ for $h\in G_{r(e)}$. We have the following commutative diagram
on $K$-theory

${K_{*}(C^{*}_{r}(G_{r(e)}))}$${K_{*}(C^{*}_{r}(G_{s(e)}))}$${K_{*}(C^{*}_{r}(G_{r(e^{\prime})}))}$${K_{*}(C^{*}_{r}(G_{s(e^{\prime})})),}$$\scriptstyle{(\theta_{e})_{*}}$$\scriptstyle{(\operatorname{Ad}u_{g})_{*}}$$\scriptstyle{(\operatorname{Ad}u_{g})_{*}}$$\scriptstyle{(\theta_{e^{\prime}})_{*}}$

where the downwards arrows are isomorphisms.

###### Proof.

The claim about the isomorphisms follows from the fact that
$G_{r(e^{\prime})}=gG_{r(e)}g^{-1}$ and similarly
$G_{s(e^{\prime})}=gG_{s(e)}g^{-1}$. Note also that
$G_{e^{\prime}}=gG_{e}g^{-1}$. For each $h\in\Sigma_{e}$, write
$h^{\prime}\in\Sigma_{e^{\prime}}$ for the representative in
$\Sigma_{e^{\prime}}$ of the coset $ghg^{-1}G_{e^{\prime}}$, and write
$h^{\prime\prime}\in G_{s(e^{\prime})}$ for the element
$(h^{\prime})^{-1}ghg^{-1}$ (note that $h^{\prime\prime}$ is in
$G_{e^{\prime}}$ and therefore $G_{s(e^{\prime})}$ because both $h^{\prime}$
and $ghg^{-1}$ are in the same coset in $G_{r(e^{\prime})}/G_{e^{\prime}}$).

First we claim that for any $k\in G_{r(e)}$ and any $h\in\Sigma_{e}$, we have
that $(gkg^{-1})^{-1}\cdot h^{\prime}=(k^{-1}\cdot h)^{\prime}$. To see this,
first note that $(gkg^{-1})^{-1}\cdot h^{\prime}$ is the representative in
$\Sigma_{e^{\prime}}$ of the coset

$gk^{-1}g^{-1}h^{\prime}G_{e^{\prime}}=gk^{-1}g^{-1}ghg^{-1}G_{e^{\prime}}=gk^{-1}hg^{-1}G_{e^{\prime}}.$

On the other hand, $(k^{-1}\cdot h)^{\prime}$ is the representative in
$\Sigma_{e^{\prime}}$ of the coset

$g(k^{-1}\cdot h)g^{-1}G_{e^{\prime}}=g(k^{-1}\cdot
h)G_{e}g^{-1}=gk^{-1}hG_{e}g^{-1}=gk^{-1}hg^{-1}G_{e^{\prime}}.$

So $(gkg^{-1})^{-1}\cdot h^{\prime}$ and $(k^{-1}\cdot h)^{\prime}$ are both
representatives in $\Sigma_{e^{\prime}}$ of the same coset, and hence they
must be equal, as claimed.

Now note that for $h\in\Sigma_{e}$, we have
$h^{\prime}=ghg^{-1}(h^{\prime\prime})^{-1}$. So for any $k\in G_{r(e)}$, we
have

$\displaystyle(h^{\prime})^{-1}(gkg^{-1})((gkg^{-1})^{-1}\cdot h^{\prime})$
$\displaystyle=(h^{\prime})^{-1}(gkg^{-1})(k^{-1}\cdot h)^{\prime}$
$\displaystyle=(h^{\prime\prime}gh^{-1}g^{-1})(gkg^{-1})[g(k^{-1}\cdot
h)g^{-1}((k^{-1}\cdot h)^{\prime\prime})^{-1}]$
$\displaystyle=h^{\prime\prime}gh^{-1}k(k^{-1}\cdot h)g^{-1}[(k^{-1}\cdot
h)^{\prime\prime}]^{-1}.$

So, writing
$\varphi\colon\mathcal{K}(\ell^{2}(\Sigma_{e}))\to\mathcal{K}(\ell^{2}(\Sigma_{e^{\prime}}))$
for the isomorphism induced by the map $h\mapsto h^{\prime}$, and writing
$u\in\mathcal{K}(\ell^{2}(\Sigma_{e^{\prime}}))\otimes
C^{*}_{r}(G_{s(e^{\prime})})$ for the unitary defined by

$u=\sum_{h\in\Sigma_{e}}e_{h^{\prime},h^{\prime}}\otimes
u_{h^{\prime\prime}},$

we have

$\displaystyle(\theta_{e^{\prime}}\circ\operatorname{Ad}u_{g})(u_{k})$
$\displaystyle=\theta_{e^{\prime}}(u_{gkg^{-1}})$
$\displaystyle=\sum_{h^{\prime}\in\Sigma_{e^{\prime}}}e_{h^{\prime},(gkg^{-1})^{-1}\cdot
h^{\prime}}\otimes u_{(h^{\prime})^{-1}(gkg^{-1})((gkg^{-1})^{-1}\cdot
h^{\prime})}$ $\displaystyle=\sum_{h\in\Sigma_{e}}e_{h^{\prime},(k^{-1}\cdot
h)^{\prime}}\otimes u_{h^{\prime\prime}gh^{-1}k(k^{-1}\cdot
h)g^{-1}[(k^{-1}\cdot h)^{\prime\prime}]^{-1}}$
$\displaystyle=u\left(\sum_{h\in\Sigma_{e}}e_{h^{\prime},(k^{-1}\cdot
h)^{\prime}}\otimes u_{g(h^{-1}k(k^{-1}\cdot h))g^{-1}}\right)u^{*}$
$\displaystyle=(\operatorname{Ad}u)((\varphi\otimes\operatorname{Ad}u_{g})(\theta_{e}(u_{k}))).$

Since $\theta_{e^{\prime}}\circ\operatorname{Ad}u_{g}$ and
$\operatorname{Ad}u\circ(\varphi\otimes\operatorname{Ad}u_{g})\circ\theta_{e}$
agree on a generating set of a dense subspace of $C^{*}_{r}(G_{r(e)})$, they
must coincide. This means that their induced maps on $K$-theory must also be
the same, and noting that $\operatorname{Ad}u$ induces the identity map in
$K$-theory as it is an inner automorphism, we get the commutative diagram in
the statement of the lemma. ∎

From now on, we implicitly use Lemma 4.7 to freely identify the groups
$K_{*}(C^{*}_{r}(G_{v}))$ for vertices $v\in E^{0}$ in the same vertex orbit,
as well as to identify the maps $(\theta_{e})_{*}$ for edges $e\in E^{1}$ in
the same edge orbit. When convenient, we will also think of $(\theta_{e})_{*}$
as a map
$\oplus_{[v]}K_{*}(C^{*}_{r}(G_{v}))\to\oplus_{[w]}K_{*}(C^{*}_{r}(G_{w}))$ by
defining $(\theta_{e})_{*}$ to be the zero map on all summands of the domain
apart from the $[r(e)]$ summand, and extending the codomain in the natural
way.

######  Theorem 4.8.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources. Suppose that the action
$G\curvearrowright\Omega$ is amenable. Then for $\alpha_{i}:=\sum_{[e]\in
G\backslash E^{1}}(\theta_{e})_{*,i}$, $i=0,1$, we have the following six-term
exact sequence

${\bigoplus_{[v]\in G\backslash
E^{0}}K_{0}(C^{*}_{r}(G_{v}))}$${\bigoplus_{[v]\in G\backslash
E^{0}}K_{0}(C^{*}_{r}(G_{v}))}$${K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${\bigoplus_{[v]\in G\backslash
E^{0}}K_{1}(C^{*}_{r}(G_{v}))}$${\bigoplus_{[v]\in G\backslash
E^{0}}K_{1}(C^{*}_{r}(G_{v})),}$$\scriptstyle{\operatorname{id}-\alpha_{0}}$$\scriptstyle{\operatorname{id}-\alpha_{1}}$

######  Remark 4.9.

We note that the action $G\curvearrowright\Omega$ is amenable if and only if
$G_{v}$ is amenable for all $v\in E^{0}$ and the action
$G\curvearrowright\partial E$ is amenable. This is because:

  * •

$G\curvearrowright\Omega$ is amenable if and only if both induced actions
$G\curvearrowright E^{0}$ and $G\curvearrowright\partial E$ are amenable, by
Proposition 3.23 and Theorem 5.15 of [9] taken together; and

  * •

$G\curvearrowright E^{0}$ is amenable if and only if $G_{v}$ is amenable for
all $v\in E^{0}$: per [3, Remarque 4.10], the amenability of
$G\curvearrowright E^{0}$ is equivalent to the nuclearity of the crossed
product $C_{0}(E^{0})\rtimes G\cong\bigoplus_{[v]\in G\backslash
E^{0}}\mathcal{K}(\ell^{2}(G\cdot v))\otimes C^{*}(G_{v})$, and this is
nuclear if and only if $G_{v}$ is amenable for all $v\in E^{0}$.

The six-term exact sequence in the theorem is essentially the one coming from
the short exact sequence of $C^{*}$-algebras

$0\to
C_{0}(E^{0})\rtimes_{\tau_{0},r}G\xrightarrow{\iota}C_{0}(\Omega)\rtimes_{\tau,r}G\to
C_{0}(\partial E)\rtimes_{\tau_{\partial},r}G\to 0,$

where $\iota\colon C_{0}(E^{0})\rtimes_{\tau_{0},r}G\to
C_{0}(\Omega)\rtimes_{\tau,r}G$ is the canonical inclusion. The following two
propositions together yield the sequence in the theorem.

######  Proposition 4.10.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources. Suppose that the action
$G\curvearrowright\Omega$ is amenable. We have the following isomorphisms in
$K$-theory:

$\bigoplus_{[v]\in G\backslash E^{0}}K_{*}(C^{*}_{r}(G_{v}))\cong
K_{*}(C_{0}(E^{0})\rtimes_{\tau_{0},r}G)\quad\text{and}\quad\bigoplus_{[v]\in
G\backslash E^{0}}K_{*}(C^{*}_{r}(G_{v}))\cong
K_{*}(C_{0}(\Omega)\rtimes_{\tau,r}G).$

Moreover, on each direct summand, the isomorphisms are induced by the
*-homomorphisms

$\displaystyle l_{v}\colon C^{*}_{r}(G_{v})$ $\displaystyle\to
C_{0}(E^{0})\rtimes_{\tau_{0},r}G$ and $\displaystyle l_{Z(v)}\colon
C^{*}_{r}(G_{v})$ $\displaystyle\to C_{0}(\Omega)\rtimes_{\tau,r}G$
$\displaystyle u_{g}$ $\displaystyle\mapsto\xi_{v}u_{g}$ $\displaystyle u_{g}$
$\displaystyle\mapsto 1_{Z(v)}u_{g}$

respectively.

###### Proof.

The claims about the first isomorphism follow from [11, Remarks 3.13 and
3.15].

For the second isomorphism, we claim that the collection of compact open sets
$\mathcal{V}=\\{Z(v):v\in E^{0}\\}$ in $\Omega$ satisfies the conditions of
Theorem 3.9. The collection $\mathcal{V}$ is $G$-invariant since the action of
$G$ on $\Omega$ satisfies $g\cdot Z(v)=Z(g\cdot v)$ for all $g\in G$ and $v\in
E^{0}$. To see that $\mathcal{V}$ generates the collection of all compact open
subsets of $\Omega$, it is enough to show that every basis set of $\Omega$ can
be written in terms of the sets in $\mathcal{V}$ using finite unions, finite
intersections and set differences; but for each $v\in E^{0}$, we have
$Z(v)\in\mathcal{V}$ and $\\{v\\}=Z(v)\setminus\cup_{e\in vE^{1}}Z(s(e))$, so
this is true. To see that $\mathcal{V}$ is independent, note that Lemma 4.5
implies that for any vertex $v\in E^{0}$, the only subset of $Z(v)$ in
$\mathcal{V}$ containing $v$ is $Z(v)$ itself. Finally, finite alignment
follows from Proposition 4.4 and the assumption that $E$ itself is finitely
aligned.

In order to apply Theorem 3.9, we also need the group $G$ to satisfy the
Baum–Connes conjecture with coefficients in $C_{0}(E^{0})$ and
$C_{0}(\Omega)$. By [11, Remark 3.13], it is enough that all vertex
stabilisers are amenable and that the transformation groupoid $\Omega\rtimes
G$ is amenable. But we know from Remark 4.9 that each vertex stabiliser is
amenable, and the amenability of $G\curvearrowright\Omega$ implies that
$\Omega\rtimes G$ is amenable by [2, Examples 2.2.14(2)]. Hence the conditions
of Theorem 3.9 are satisfied, and the claims about the second isomorphism
follow directly from that theorem. ∎

######  Proposition 4.11.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources. Write $\alpha_{i}:=\sum_{[e]\in G\backslash
E^{1}}(\theta_{e})_{*,i}$ for $i=0,1$, and write $\iota$ for the canonical
inclusion $C_{0}(E^{0})\rtimes_{\tau_{0},r}G\to
C_{0}(\Omega)\rtimes_{\tau,r}G$. We have the following commutative diagram for
$i=0,1$:

${\bigoplus_{[v]\in G\backslash
E^{0}}K_{i}(C^{*}_{r}(G_{v}))}$${\bigoplus_{[w]\in G\backslash
E^{0}}K_{i}(C^{*}_{r}(G_{w}))}$${K_{i}(C_{0}(E^{0})\rtimes_{\tau_{0},r}G)}$${K_{i}(C_{0}(\Omega)\rtimes_{\tau,r}G).}$$\scriptstyle{\operatorname{id}-\alpha_{i}}$$\scriptstyle{\bigoplus_{[v]}(l_{v})_{*,i}}$$\scriptstyle{\bigoplus_{[w]}(l_{Z(w)})_{*,i}}$$\scriptstyle{\iota_{*,i}}$

To prove Proposition 4.11, we need some notation and two lemmas.

######  Notation 4.12.

For each $e\in E^{1}$, write

$1_{[e]}:=\sum_{e^{\prime}\in G_{r(e)}\cdot e}1_{Z(s(e^{\prime}))}\in
C_{0}(\Omega)$

and write $l_{[e]}$ for the map

$C^{*}_{r}(G_{r(e)})\to C_{0}(\Omega)\rtimes_{\tau,r}G,\quad u_{g}\mapsto
1_{[e]}u_{g}.$

Write

$C_{[e]}:=C^{*}(\\{1_{Z(s(e^{\prime}))}:e^{\prime}\in G_{r(e)}\cdot
e\\})\rtimes_{\tau|,r}G_{r(e)}\subseteq C_{0}(\Omega)\rtimes_{\tau,r}G.$

######  Lemma 4.13.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources, and fix $e\in E^{1}$. The element

$u_{[e]}=\sum_{h,k\in\Sigma_{e}}e_{h,k}\otimes u_{kh^{-1}}1_{Z(s(h\cdot
e))}\in\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes C_{[e]}$

is a self-adjoint unitary in $\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes
C_{[e]}$.

###### Proof.

To see that $u_{[e]}$ is self-adjoint, note that

$\displaystyle u_{[e]}^{*}$
$\displaystyle=\sum_{h,k\in\Sigma_{e}}e_{k,h}\otimes 1_{Z(s(h\cdot
e))}u^{*}_{kh^{-1}}$ $\displaystyle=\sum_{h,k\in\Sigma_{e}}e_{k,h}\otimes
u^{*}_{kh^{-1}}1_{Z(s(k\cdot e))}$
$\displaystyle=\sum_{h,k\in\Sigma_{e}}e_{k,h}\otimes u_{hk^{-1}}1_{Z(s(k\cdot
e))}$ $\displaystyle=u_{[e]}.$

To see that $u_{[e]}$ is unitary, first note that $C_{[e]}$ is unital with
identity element $1_{[e]}$. Now

$\displaystyle u_{[e]}u_{[e]}^{*}$
$\displaystyle=\left(\sum_{h,k\in\Sigma_{e}}e_{h,k}\otimes
u_{kh^{-1}}1_{Z(s(h\cdot
e))}\right)\left(\sum_{k,l\in\Sigma_{e}}e_{k,l}\otimes 1_{Z(s(l\cdot
e))}u^{*}_{kl^{-1}}\right)$
$\displaystyle=\sum_{h,l\in\Sigma_{e}}e_{h,l}\otimes\left(\sum_{k\in\Sigma_{e}}u_{kh^{-1}}1_{Z(s(h\cdot
e))}1_{Z(s(l\cdot e))}u^{*}_{kl^{-1}}\right).$

Note that the projections $\\{1_{Z(s(h\cdot e))}:h\in\Sigma_{e}\\}$ are
pairwise orthogonal. So continuing the calculation, we use covariance to get

$\displaystyle u_{[e]}u_{[e]}^{*}$
$\displaystyle=\sum_{h\in\Sigma_{e}}e_{h,h}\otimes\left(\sum_{k\in\Sigma_{e}}u_{kh^{-1}}1_{Z(s(h\cdot
e))}u^{*}_{kh^{-1}}\right)$
$\displaystyle=\sum_{h\in\Sigma_{e}}e_{h,h}\otimes\left(\sum_{k\in\Sigma_{e}}1_{Z(s(k\cdot
e))}u_{kh^{-1}}u^{*}_{kh^{-1}}\right)$
$\displaystyle=\sum_{h\in\Sigma_{e}}e_{h,h}\otimes\left(\sum_{k\in\Sigma_{e}}1_{Z(s(k\cdot
e))}\right)$ $\displaystyle=\sum_{h\in\Sigma_{e}}e_{h,h}\otimes 1_{[e]}$
$\displaystyle=1_{\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes C_{[e]}}.$

Since $u_{[e]}$ is self-adjoint, we also have that
$u_{[e]}^{*}u_{[e]}=1_{\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes C_{[e]}}$. ∎

######  Lemma 4.14.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources. For any $e\in E^{1}$, we have

$(l_{Z(s(e))})_{*}\circ(\theta_{e})_{*}=(l_{[e]})_{*}.$

###### Proof.

Fix $e\in E^{1}$. Write $l^{\prime}_{[e]}\colon C^{*}_{r}(G_{r(e)})\to
C_{[e]}$ for the map defined by $u_{g}\mapsto 1_{[e]}u_{g}$ (this is just
$l_{[e]}$ with codomain restricted to $C_{[e]}$), and write $\iota_{[e]}$ for
the inclusion $C_{[e]}\hookrightarrow C_{0}(\Omega)\rtimes_{r}G$. We need to
show that

$(l_{Z(s(e))})_{*}\circ(\theta_{e})_{*}=(\iota_{[e]})_{*}\circ(l^{\prime}_{[e]})_{*}.$

Write $\varphi_{e}$ for the map

$\varphi_{e}\colon C_{[e]}\to\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes
C_{[e]},\qquad a\mapsto u_{[e]}(e_{1,1}\otimes a)u_{[e]}^{*},$

where $u_{[e]}$ is the self-adjoint unitary from Lemma 4.13 and where
$1\in\Sigma_{e}$ denotes the identity of $G_{r(e)}$. Since $\varphi_{e}$ is
the composition of $a\mapsto e_{1,1}\otimes a$ and an inner automorphism, both
of which induce the identity map on $K$-theory, the map $\varphi_{e}$ itself
induces the identity map

$(\varphi_{e})_{*}=\operatorname{id}\colon K_{*}(C_{[e]})\to K_{*}(C_{[e]}).$

So it is enough to show that

(4.2)
$(l_{Z(s(e))})_{*}\circ(\theta_{e})_{*}=(\iota_{[e]})_{*}\circ(\varphi_{e})_{*}\circ(l^{\prime}_{[e]})_{*}.$

To do this, we show that the following diagram is commutative; Equation (4.2)
then follows from the commutativity of the induced diagram on $K$-theory:

(4.3) ${C^{*}_{r}(G_{r(e)})}$${\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes
C^{*}_{r}(G_{s(e)})}$${C_{[e]}}$${\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes
C_{[e]}}$${\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes(C_{0}(\Omega)\rtimes_{r}G).}$$\scriptstyle{\theta_{e}}$$\scriptstyle{l^{\prime}_{[e]}}$$\scriptstyle{\operatorname{id}\otimes
l_{Z(s(e))}}$$\scriptstyle{\varphi_{e}}$$\scriptstyle{\operatorname{id}\otimes\iota_{[e]}}$

Consider the composition
$(\operatorname{id}\otimes\iota_{[e]})\circ\varphi_{e}\circ
l^{\prime}_{[e]}\colon
C^{*}_{r}(G_{r(e)})\to\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes(C_{0}(\Omega)\rtimes_{r}G)$.
For $g\in G_{r(e)}$, we have

$\displaystyle((\operatorname{id}\otimes\iota_{[e]})\circ\varphi_{e}\circ
l^{\prime}_{[e]})(u_{g})$
$\displaystyle=((\operatorname{id}\otimes\iota_{[e]})\circ\varphi_{e})(1_{[e]}u_{g})$
$\displaystyle=(\operatorname{id}\otimes\iota_{[e]})(u_{[e]}(e_{1,1}\otimes
1_{[e]}u_{g})u_{[e]}^{*})$
$\displaystyle=\sum_{h,k\in\Sigma_{e}}e_{h,k}\otimes
1_{Z(s(e))}u_{h^{-1}gk}1_{Z(s(e))}$
$\displaystyle=\sum_{h,k\in\Sigma_{e}}e_{h,k}\otimes
1_{Z(s(e))}1_{Z(s(h^{-1}gk\cdot e))}u_{h^{-1}gk}.$

Note that $1_{Z(s(e))}1_{Z(s(h^{-1}gk\cdot e))}$ is non-zero if and only if
$h^{-1}gk\in G_{e}$, in which case it is equal to $1_{Z(s(e))}$. But
$h^{-1}gk\in G_{e}$ if and only if $k=g^{-1}\cdot h$ as elements of
$\Sigma_{e}$, so continuing the calculation, we have

$\displaystyle((\operatorname{id}\otimes\iota_{[e]})\circ\varphi_{e}\circ
l^{\prime}_{[e]})(u_{g})$ $\displaystyle=\sum_{h\in\Sigma_{e}}e_{h,g^{-1}\cdot
h}\otimes 1_{Z(s(e))}u_{h^{-1}g(g^{-1}\cdot h)}$
$\displaystyle=((\operatorname{id}\otimes
l_{Z(s(e))})\circ\theta_{e})(u_{g}).$

Since the maps $(\operatorname{id}\otimes\iota_{[e]})\circ\varphi_{e}\circ
l^{\prime}_{[e]}$ and $(\operatorname{id}\otimes l_{Z(s(e))})\circ\theta_{e}$
agree on a generating set of a dense sub-$C^{*}$-algebra of
$C^{*}_{r}(G_{v})$, they are equal, which proves that the diagram Equation
(4.3) is commutative. ∎

We are now ready to prove Proposition 4.11, and then complete the proof of
Theorem 4.8.

###### Proof of Proposition 4.11.

Fix $v\in E^{0}$. Note that

$1_{Z(v)}=\xi_{v}+\sum_{[e]\in G_{v}\backslash vE^{1}}1_{[e]},$

so the *-homomorphisms $\iota\circ l_{v}$ and $l_{[e]}$, $[e]\in
G_{v}\backslash vE^{1}$ from $C^{*}_{r}(G_{v})\to
C_{0}(\Omega)\rtimes_{\tau,r}G$ are mutually orthogonal. Thus the induced
homomorphisms in $K$-theory satisfy

$\iota_{*}\circ(l_{v})_{*}=(\iota\circ l_{v})_{*}=(l_{Z(v)})_{*}-\sum_{[e]\in
G_{v}\backslash vE^{1}}(l_{[e]})_{*}.$

Now, Proposition 4.14 says that for all $e\in vE^{1}$, we have

$(l_{Z(s(e))})_{*}\circ(\theta_{e})_{*}=(l_{[e]})_{*}.$

Summing over a set of representatives of $G_{v}\backslash vE^{1}$, we get that

$\left(\bigoplus_{[w]\in G\backslash
E^{0}}(l_{Z(w)})_{*}\right)\circ\left(\sum_{[e]\in G_{v}\backslash
vE^{1}}(\theta_{e})_{*}\right)=\sum_{[e]\in G_{v}\backslash
vE^{1}}(l_{[e]})_{*}.$

Note also that, if we write $\iota_{v}$ for the canonical inclusion of
$K_{*}(C^{*}_{r}(G_{v}))$ into the direct sum
$\oplus_{[w]}K_{*}(C^{*}_{r}(G_{w}))$, we trivially have that

$\left(\bigoplus_{[w]\in G\backslash
E^{0}}(l_{Z(w)})_{*}\right)\circ\iota_{v}=(l_{Z(v)})_{*}.$

Putting the above two equations together gives that

$\left(\bigoplus_{[w]\in G\backslash
E^{0}}(l_{Z(w)})_{*}\right)\circ\left(\iota_{v}-\sum_{[e]\in G_{v}\backslash
vE^{1}}(\theta_{e})_{*}\right)=(l_{Z(v)})_{*}-\sum_{[e]\in G_{v}\backslash
vE^{1}}(l_{[e]})_{*}=\iota_{*}\circ(l_{v})_{*}.$

Note that $\oplus_{[v]\in G\backslash E^{0}}\iota_{v}$ is the identity map on
$\oplus_{[v]\in G\backslash E^{0}}K_{*}(C^{*}_{r}(G_{v}))$, so taking a direct
sum of the above equation over $[v]\in G\backslash E^{0}$ (now allowing $v$ to
vary) gives

$\left(\bigoplus_{[w]\in G\backslash
E^{0}}(l_{Z(w)})_{*}\right)\circ\left(\operatorname{id}-\bigoplus_{[v]\in
G\backslash E^{0}}\sum_{[e]\in G_{v}\backslash
vE^{1}}(\theta_{e})_{*}\right)=\iota_{*}\circ\left(\bigoplus_{[v]\in
G\backslash E^{0}}(l_{v})_{*}\right).$

Now all that remains to be shown is that, for $i=0,1$,

$\bigoplus_{[v]\in G\backslash E^{0}}\sum_{[e]\in G_{v}\backslash
vE^{1}}(\theta_{e})_{*,i}=\alpha_{i}=\sum_{[e]\in G\backslash
E^{1}}(\theta_{e})_{*,i},$

which follows from the fact that $G_{v}\backslash vE^{1}\cong\\{[e]\in
G\backslash E^{1}:r(e)\in G\cdot v\\}$. ∎

###### Proof of Theorem 4.8.

Since $\partial E$ is a closed subspace of $\Omega$ and $E^{0}$ is its
complement, we have a short exact sequence

$0\to C_{0}(E^{0})\to C_{0}(\Omega)\to C_{0}(\partial E)\to 0,$

which induces a short exact sequence of full crossed products

$0\to
C_{0}(E^{0})\rtimes_{\tau_{0}}G\xrightarrow{\iota}C_{0}(\Omega)\rtimes_{\tau}G\to
C_{0}(\partial E)\rtimes_{\tau_{\partial}}G\to 0.$

Now note the following. By Remark 4.9, amenability of the action
$G\curvearrowright\Omega$ implies that the actions $G\curvearrowright E^{0}$
and $G\curvearrowright\partial E$ are amenable. Then [9, Theorem 5.15] gives
that all three actions $\tau_{0}$, $\tau$ and $\tau_{\partial}$ of $G$ on
$C_{0}(E^{0})$, $C_{0}(\Omega)$ and $C_{0}(\partial E)$ respectively are
amenable. Finally, since $G$ is discrete, amenability of these actions implies
that the corresponding full crossed products are isomorphic to the respective
reduced crossed products [3, Proposition 4.8]. Altogether, this implies that
the sequence of reduced crossed products

$0\to
C_{0}(E^{0})\rtimes_{\tau_{0},r}G\xrightarrow{\iota}C_{0}(\Omega)\rtimes_{\tau,r}G\to
C_{0}(\partial E)\rtimes_{\tau_{\partial},r}G\to 0$

is also exact, where $\iota\colon C_{0}(E^{0})\rtimes_{r}G\to
C_{0}(\Omega)\rtimes_{r}G$ is the canonical inclusion. This gives the
following six-term exact sequence in $K$-theory:

${K_{0}(C_{0}(E^{0})\rtimes_{\tau_{0},r}G)}$${K_{0}(C_{0}(\Omega)\rtimes_{\tau,r}G)}$${K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\Omega)\rtimes_{\tau,r}G)}$${K_{1}(C_{0}(E^{0})\rtimes_{\tau_{0},r}G)}$$\scriptstyle{\iota_{*,0}}$$\scriptstyle{\iota_{*,1}}$

Then Propositions 4.10 and 4.11 together imply the result. ∎

######  Remark 4.15.

We note that the amenability assumptions in Theorem 4.8 are probably stronger
than necessary for the statement. All that is strictly necessary for the proof
is that $G$ satisfies the Baum–Connes conjecture with coefficients in
$C_{0}(E^{0})$ and $C_{0}(\Omega)$ (so that Theorem 3.9 can be applied), and
that the sequence of crossed products

$0\to
C_{0}(E^{0})\rtimes_{\tau_{0},r}G\xrightarrow{\iota}C_{0}(\Omega)\rtimes_{\tau,r}G\to
C_{0}(\partial E)\rtimes_{\tau_{\partial},r}G\to 0$

is exact.

###  4.3. Example: trivial vertex stabilisers

In this subsection and the next, we provide more specialised $K$-theory
formulas for the crossed product $C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G$ in the case where all vertex stabilisers are
trivial (equivalently where $G$ acts freely on $E$), and in the case where all
vertex stabilisers are infinite cyclic.

######  Theorem 4.16.

Let $G$ be a countable discrete group acting freely on a row-finite, finitely
aligned multitree $E$ with no sources. Suppose that the action
$G\curvearrowright\Omega$ is amenable. Write $\Gamma:=G\backslash E$ for the
quotient directed graph, and let $A\in M_{\Gamma^{0}}(\mathbb{Z})$ be the
adjacency matrix of $\Gamma$ (defined by $A_{v,w}=\\#v\Gamma^{1}w$). The
$K$-theory of the reduced crossed product $C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G$ is given by

$K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)\cong\operatorname{coker}(1-A^{T})\quad\text{and}\quad
K_{1}(C_{0}(\partial E)\rtimes_{\tau_{\partial},r}G)\cong\ker(1-A^{T}),$

where we consider $1-A^{T}$ to be a homomorphism
$\mathbb{Z}[\Gamma^{0}]\to\mathbb{Z}[\Gamma^{0}]$.

###### Proof.

Since $G$ acts freely on $E$, all stabiliser subgroups are trivial. Note that
the (reduced) $C^{*}$-algebra of the trivial group is simply $\mathbb{C}$, and
so for all $v\in E^{0}$, we have $C^{*}_{r}(G_{v})\cong\mathbb{C}$. So
(writing $1_{v}$ for the unique element in $G_{v}$)
$K_{0}(C^{*}_{r}(G_{v}))=\mathbb{Z}[1_{v}]$ and $K_{1}(C^{*}_{r}(G_{v}))=0$
for all $v\in E^{0}$. So applying Theorem 4.8, we get the six-term exact
sequence

${\bigoplus_{[v]\in G\backslash E^{0}}\mathbb{Z}[1_{v}]}$${\bigoplus_{[v]\in
G\backslash E^{0}}\mathbb{Z}[1_{v}]}$${K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${0}$${0,}$$\scriptstyle{\operatorname{id}-\alpha_{0}}$

where

$\alpha_{0}=\sum_{[e]\in G\backslash E^{1}}(\theta_{e})_{*,0}.$

For each $e\in E^{1}$, the map $\theta_{e}$ sends $1_{r(e)}$ to $1_{s(e)}$,
and so the induced map on $K_{0}$ satisfies
$(\theta_{e})_{*,0}([1_{r(e)}])=[1_{s(e)}]$. From this, it is easy to see that
$\alpha_{0}$ is exactly (left multiplication by the matrix) $A^{T}$. ∎

######  Remark 4.17.

We can recover the well-known formula for the $K$-theory of directed graph
$C^{*}$-algebras [21] (at least in the row-finite, no sources case) from the
above theorem, using that a directed graph algebra is Morita equivalent to the
crossed product of a free group acting on the boundary of the covering tree of
the graph [15, Corollary 4.14]. The underlying action of the group on the
covering tree is free [15, Lemma 4.10], which implies that all vertex
stabilisers are trivial; Proposition 6.11 then gives that the action on the
boundary is also amenable. Finally, Remark 4.9 implies that the action on
$\Omega$ is amenable, thereby allowing us to apply Theorem 4.16.

###  4.4. Example: infinite cyclic vertex stabilisers

We now move to the case where all vertex stabilisers (and hence all edge
stabilisers) are isomorphic to $\mathbb{Z}$. First we introduce some notation.

######  Notation 4.18.

Let $G$ be a discrete group acting on a row-finite, finitely aligned multitree
$E$ with no sources. For any path (finite or infinite) $\alpha$ in $E$, write
$G_{\alpha}$ for the subgroup of $G$ which fixes $\alpha$. We write
$\Gamma:=G\backslash E$ for the quotient directed graph. For each vertex
$v\in\Gamma^{0}$ we choose a lift $v^{\prime}\in E^{0}$ of $v$, and for each
edge $e\in\Gamma^{1}$ we choose a lift $e^{\prime}\in E^{1}$ of $e$. For each
$v\in\Gamma^{0}$ and $e\in\Gamma^{1}$ we define groups $G_{v}$ and $G_{e}$ by
$G_{v}:=G_{v^{\prime}}$ and $G_{e}:=G_{e^{\prime}}$.

For each $e\in\Gamma^{1}$, the group $G_{e}$ embeds into the group $G_{r(e)}$
in the following way. Write $v=r(e)$, and let $v^{\prime}\in E^{0}$ be the
chosen lift of $v$. The vertex $r(e^{\prime})\in E^{0}$ is also a lift of $v$,
and so will be in the same vertex orbit as $v$. In particular, there is some
group element $g\in G$ such that $g\cdot r(e^{\prime})=v^{\prime}$. Then
$G_{v^{\prime}}=gG_{r(e^{\prime})}g^{-1}$, and so we can define an injective
homomorphism $G_{e}\hookrightarrow G_{r(e)}$ by $h\mapsto ghg^{-1}$. Replacing
‘$r$’s with ‘$s$’s in the above argument shows that $G_{e}$ also embeds into
$G_{s(e)}$ in a similar way. We will often identify $G_{e}$ with its image in
$G_{r(e)}$ or $G_{s(e)}$ under the respective embeddings.

We now introduce additional notation for the special situation where for each
$v^{\prime}\in E^{0}$, the group $G_{v^{\prime}}$ is infinite cyclic. For
$e\in\Gamma^{1}$, we have by [Section 2.2(3)] that $G_{e}$ is a subgroup of
$G_{r(e)}$ of finite index, and hence is also infinite cyclic. For each
$v\in\Gamma^{0}$ choose a generator $1_{v^{\prime}}$ of
$G_{v^{\prime}}\cong\mathbb{Z}$ (where $v^{\prime}$ is the chosen lift of
$v$). Similarly, for each edge $e\in\Gamma^{1}$ we choose a generator
$1_{e^{\prime}}$ of $G_{e^{\prime}}\cong\mathbb{Z}$ (where $e^{\prime}$ is the
chosen lift of $e$).

Finally, to each $e\in\Gamma^{1}$ we associate integers
$\omega_{e},\omega_{\overline{e}}$ in the following way: we denote by
$\omega_{e}$ the element of $\mathbb{Z}$ such that the embedding of $G_{e}$
into $G_{r(e)}$ sends $1_{e^{\prime}}$ to $\omega_{e}1_{r(e)^{\prime}}$, and
similarly we denote by $\omega_{\overline{e}}$ the element of $\mathbb{Z}$
such that the embedding of $G_{e}$ into $G_{s(e)}$ sends $1_{e^{\prime}}$ to
$\omega_{\overline{e}}1_{s(e)^{\prime}}$.

######  Theorem 4.19.

Let $G$ be a countable discrete group acting on a row-finite, finitely aligned
multitree $E$ with no sources. Suppose that for each $v\in E^{0}$, the
stabiliser $G_{v}\leq G$ is isomorphic to $\mathbb{Z}$. Suppose further that
the action $G\curvearrowright\Omega$ is amenable. Define maps
$A_{0},A_{1}\colon\mathbb{Z}[\Gamma^{0}]\to\mathbb{Z}[\Gamma^{0}]$ by

$A_{0}(v)=\sum_{e\in v\Gamma^{1}}\lvert\omega_{e}\rvert
s(e)\qquad\text{and}\qquad A_{1}(v)=\sum_{e\in
v\Gamma^{1}}\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}s(e)$

for $v\in\Gamma^{0}$. We have

$K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)\cong\operatorname{coker}(1-A_{0})\oplus\ker(1-A_{1})$

and

$K_{1}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)\cong\operatorname{coker}(1-A_{1})\oplus\ker(1-A_{0}).$

###### Proof.

Recall that $K_{0}(C^{*}_{r}(\mathbb{Z}))=\mathbb{Z}[u_{0}]_{0}$ and
$K_{1}(C^{*}_{r}(\mathbb{Z}))=\mathbb{Z}[u_{1}]_{1}$. For $v\in E^{0}$ and
$n\in\mathbb{Z}$, write $u^{(v)}_{n}\in C^{*}_{r}(G_{v})$ for the unitary
representation of $n\in\mathbb{Z}\cong G_{v}$; we omit the superscript ‘$(v)$’
when it is clear what the vertex is. So for all $v\in E^{0}$, we have

$K_{0}(C^{*}_{r}(G_{v}))=\mathbb{Z}[u^{(v)}_{0}]_{0},\qquad
K_{1}(C^{*}_{r}(G_{v}))=\mathbb{Z}[u^{(v)}_{1}]_{1}.$

From Theorem 4.8, we get the six-term exact sequence

${\bigoplus_{[v]\in G\backslash
E^{0}}\mathbb{Z}[u^{(v)}_{0}]_{0}}$${\bigoplus_{[v]\in G\backslash
E^{0}}\mathbb{Z}[u^{(v)}_{0}]_{0}}$${K_{0}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\partial
E)\rtimes_{\tau_{\partial},r}G)}$${\bigoplus_{[v]\in G\backslash
E^{0}}\mathbb{Z}[u^{(v)}_{1}]_{1}}$${\bigoplus_{[v]\in G\backslash
E^{0}}\mathbb{Z}[u^{(v)}_{1}]_{1},}$$\scriptstyle{\operatorname{id}-\alpha_{0}}$$\scriptstyle{\operatorname{id}-\alpha_{1}}$

where

$\alpha_{i}=\sum_{[e]\in G\backslash E^{1}}(\theta_{e})_{*,i}.$

Now $\theta_{e}$ behaves in the following way. Fix $e\in E^{1}$. We know that
$G_{e}$ embeds into $G_{r(e)}$ as the subgroup $\omega_{e}G_{r(e)}$, so we can
choose $\Sigma_{e}$ to be the set $\\{0,1,\dots,\lvert\omega_{e}\rvert-1\\}$
(where we have identified $G_{r(e)}$ with $\mathbb{Z}$). Then the map

$\theta_{e}\colon
C^{*}_{r}(G_{r(e)})\to\mathcal{K}(\ell^{2}(\Sigma_{e}))\otimes
C^{*}_{r}(G_{s(e)})\cong M_{\lvert\omega_{e}\rvert}(C^{*}_{r}(G_{s(e)}))$

satisfies

$\theta_{e}(u^{(r(e))}_{0})=\sum_{h=0}^{\lvert\omega_{e}\rvert-1}e_{h,h}\otimes
u^{(s(e))}_{0}=\operatorname{diag}(u^{(s(e))}_{0},\dots,u^{(s(e))}_{0})$

and

$\theta_{e}(u^{(r(e))}_{1})=\sum_{h=0}^{\lvert\omega_{e}\rvert-1}e_{h,(-1)\cdot
h}\otimes u^{(s(e))}_{-h+1+((-1)\cdot
h)}=e_{0,\lvert\omega_{e}\rvert-1}\otimes u^{(s(e))}_{\lvert\omega_{e}\rvert
1_{r(e)}}+\sum_{h=1}^{\lvert\omega_{e}\rvert-1}e_{h,h-1}\otimes
u^{(s(e))}_{0}.$

Now note that $\lvert\omega_{e}\rvert
1_{r(e)}=\frac{\lvert\omega_{e}\rvert}{\omega_{e}}1_{e}=\operatorname{sgn}(\omega_{e})1_{e}=\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}1_{s(e)}$,
so $u^{(s(e))}_{\lvert\omega_{e}\rvert
1_{r(e)}}=u^{(s(e))}_{\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}}$.
This means that

$\theta_{e}(u^{(r(e))}_{1})=e_{0,\lvert\omega_{e}\rvert-1}\otimes
u^{(s(e))}_{\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}}+\sum_{h=1}^{\lvert\omega_{e}\rvert-1}e_{h,h-1}\otimes
u^{(s(e))}_{0}=\begin{pmatrix}0&u^{(s(e))}_{\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}}\\\
\operatorname{diag}(u^{(s(e))}_{0},\dots,u^{(s(e))}_{0})&0\end{pmatrix}$

in block matrix form. But this matrix is homotopy equivalent to the diagonal
matrix
$\operatorname{diag}(u^{(s(e))}_{\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}},u^{(s(e))}_{0},\dots,u^{(s(e))}_{0})$.

To summarise, we have

$\theta_{e}(u_{0})=\operatorname{diag}(u_{0},\dots,u_{0})\qquad\text{and}\qquad\theta_{e}(u_{1})\sim_{h}\operatorname{diag}(u_{\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}},u_{0},\dots,u_{0}),$

so the induced maps on $K$-theory are given by

$\displaystyle(\theta_{e})_{*,0}\colon\mathbb{Z}[u_{0}]_{0}$
$\displaystyle\to\mathbb{Z}[u_{0}]_{0}$
$\displaystyle(\theta_{e})_{*,1}\colon\mathbb{Z}[u_{1}]_{1}$
$\displaystyle\to\mathbb{Z}[u_{1}]_{1}$ $\displaystyle[u_{0}]_{0}$
$\displaystyle\mapsto\lvert\omega_{e}\rvert[u_{0}]_{0},$
$\displaystyle[u_{1}]_{1}$
$\displaystyle\mapsto[u_{\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}}]_{1}=\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}[u_{1}]_{1},$

and hence $(\theta_{e})_{*,0}$ is multiplication by $\lvert\omega_{e}\rvert$,
and $(\theta_{e})_{*,1}$ is multiplication by
$\operatorname{sgn}(\omega_{e})\omega_{\overline{e}}$. The theorem then
follows from the commutativity of the diagram

${\bigoplus_{[v]\in G\backslash E^{0}}K_{i}(C^{*}(G_{v}))}$${\bigoplus_{[v]\in
G\backslash
E^{0}}K_{i}(C^{*}(G_{v}))}$${\mathbb{Z}[\Gamma^{0}]}$${\mathbb{Z}[\Gamma^{0}]}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{A_{i}}$

for $i=0,1$. ∎

##  5\. Group actions on undirected trees

We can also use Theorem 4.8 to study crossed products coming from actions on
the boundaries of undirected trees. Such actions arise naturally for example
in the study of graph-of-groups $C^{*}$-algebras (see [8], in particular
Theorem 4.1).

In Section 5.1 we discuss the notion of a dual graph, and how it brings group
actions on undirected trees into our setting of group actions on multitrees.
Then, in Section 5.2, we explain how Theorem 4.8 can be applied to yield a
six-term exact sequence for the $K$-theory of crossed products induced by a
group action on an undirected tree.

###  5.1. Dual graphs

In this subsection, we explain how we can use a dual graph construction to
associate to any undirected tree a suitable multitree. We mirror the
construction of the dual graph for directed graphs (see, for example, [6, Page
313]), except that we only consider reduced paths.

######  Definition 5.1.

Let $\Gamma$ be a undirected graph. The dual graph of $\Gamma$ is the directed
graph
$\widehat{\Gamma}=(\widehat{\Gamma}^{0},\widehat{\Gamma}^{1},\widehat{r},\widehat{s})$
defined by

$\widehat{\Gamma}^{0}=\Gamma^{1},\quad\widehat{\Gamma}^{1}=\Gamma^{2}_{r},\quad\widehat{r}(ef)=e,\quad\widehat{s}(ef)=f,\quad
ef\in\Gamma^{2}_{r}.$

######  Example 5.2.

Below is an example of an undirected tree (on the left), along with its dual
multitree (on the right).

$\cdots$$\cdots$$\cdots$$\cdots$$e$$\overline{e}$ $e$$\overline{e}$

$\cdots$

$\cdots$

$\cdots$

$\cdots$

Note that there is a natural identification of paths (finite or infinite) in
$\widehat{\Gamma}$ with reduced paths (finite or infinite) of length at least
one in $\Gamma$, given by viewing vertices in $\widehat{\Gamma}$ as edges in
$\Gamma$. We will use this identification in the proof of the following
proposition, which explains why this dual graph construction is useful.

######  Proposition 5.3.

Let $T$ be a locally finite nonsingular undirected tree. Its dual graph
$\widehat{T}$ is a row-finite, finitely aligned multitree with no sources, and
$\partial T\cong\partial\widehat{T}$.

Suppose further that a discrete group $G$ acts on $T$. Then $G$ acts naturally
on $\partial\widehat{T}$ and $G\curvearrowright\partial T\cong
G\curvearrowright\partial\widehat{T}$.

###### Proof.

First we show that $\widehat{T}$ is a multitree. Take two vertices
$e,f\in\widehat{T}^{0}$, and suppose for a contradiction that there are two
distinct paths $\lambda,\mu\in e\widehat{T}^{*}f$. This implies that there are
two distinct reduced paths in $T$ from $r(f)$ to $s(e)$, contradicting that
$T$ is a tree. Hence there is at most one directed path between any two
vertices in $\widehat{T}^{0}$, so $\widehat{T}$ is a multitree.

To see that $\widehat{T}$ is row-finite and has no sources, note that for any
vertex $e\in\widehat{T}^{0}$, the set
$\widehat{r}^{-1}(e)\subseteq\widehat{T}^{1}$ is in a one-to-one
correspondence with the set $s(e)T^{1}\setminus\\{\overline{e}\\}$. In
particular, we have that $\lvert\widehat{r}^{-1}(e)\rvert=\lvert
s(e)T^{1}\rvert-1$. So the assumption that $T$ is locally finite and
nonsingular implies that $\widehat{T}$ is row-finite and has no sources.

To see that $\widehat{T}$ is finitely aligned, fix $e,f\in\widehat{T}^{0}$. We
need to show that the set of minimal common upper bounds for $e$ and $f$ is
finite. If $e\leq f$ (respectively $f\leq e$) then the only minimal common
upper bound for $e$ and $f$ is $f$ (respectively $e$), so we are done. So
suppose that $e$ and $f$ are not comparable but have common upper bounds. Take
some common upper bound $x$ of $e$ and $f$, so there are paths $\lambda\in
e\widehat{T}^{*}x$ and $\mu\in f\widehat{T}^{*}x$, which we will view as
reduced paths in $T$. Write $\nu$ for the longest path such that
$\lambda=\alpha\nu$ and $\mu=\beta\nu$ for some paths $\alpha$ and $\beta$.
Then, writing $\overline{\beta}$ for the reversal of the path $\beta$, we have
that $\alpha\overline{\beta}$ is the (unique) reduced path in $T$ from $r(f)$
to $r(e)$, and note that $r(\nu)=s(\alpha)$ lies on this path. Write $y$ for
the range-most edge in $\nu$. Then $y$ is a common upper bound of $e$ and $f$,
and we have $y\leq x$. This shows that any common upper bound of $e$ and $f$
lies above some common upper bound $y$ with $r(y)\in T^{0}$ lying on the
reduced path in $T$ from $r(f)$ to $r(e)$. But this path is finite, and since
$T$ is locally finite, there are only finitely many such $y$’s. Hence $e$ and
$f$ have only finitely many minimal common upper bounds.

We now show that $\partial T\cong\partial\widehat{T}$. First notice that the
sets $T^{\infty}_{r}$ and $\widehat{T}^{\infty}$ are isomorphic under the
natural identification. This isomorphism respects shift-tail equivalence, and
so the boundaries $\partial T$ and $\partial\widehat{T}$ are isomorphic as
sets. Moreover, for any edge $e\in T^{1}$, the cylinder set
$Z_{\partial}(e)\subseteq\partial T$ is sent by the isomorphism to the
cylinder set $Z_{\partial}(e)\subseteq\partial\widehat{T}$, where $e$ is
considered as a vertex of $\widehat{T}$. Thus $\partial
T\cong\partial\widehat{T}$ as topological spaces.

Finally, suppose that a discrete group $G$ acts on $T$. This induces an action
of $G$ on $\widehat{T}$ in the natural way, which in turn induces an action of
$G$ on $\partial\widehat{T}$, and the isomorphism $\partial
T\cong\partial\widehat{T}$ described above intertwines the actions of $G$ on
$\partial T$ and $\partial\widehat{T}$, which implies that
$G\curvearrowright\partial T\cong G\curvearrowright\partial\widehat{T}$, as
required. ∎

###  5.2. Group actions on boundaries of undirected trees

In this subsection we apply Theorem 4.8 to arrive at the following result for
groups acting on the boundaries of undirected trees. Note that we write
$G_{ef}$ for the stabiliser subgroup of $G$ at $ef\in
T^{2}_{r}=\widehat{T}^{1}$, and the map $\theta_{ef}$ is defined as in
Notation 4.6, where $ef$ is considered as an edge in the dual multitree
$\widehat{T}$.

######  Theorem 5.4.

Let $G$ be a countable discrete group acting on a locally finite nonsingular
undirected tree $T$. Suppose that $G_{v}$ is amenable for all $v\in T^{0}$.
Then for $\widehat{\alpha}_{i}=\sum_{[ef]\in G\backslash
T^{2}_{r}}(\theta_{ef})_{*,i}$ we have the following six-term exact sequence

${\bigoplus_{[e]\in G\backslash
T^{1}}K_{0}(C^{*}_{r}(G_{e}))}$${\bigoplus_{[ef]\in G\backslash
T^{2}_{r}}K_{0}(C^{*}_{r}(G_{ef}))}$${K_{0}(C_{0}(\partial
T)\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\partial
T)\rtimes_{\tau_{\partial},r}G)}$${\bigoplus_{[ef]\in G\backslash
T^{2}_{r}}K_{1}(C^{*}_{r}(G_{ef}))}$${\bigoplus_{[e]\in G\backslash
T^{1}}K_{1}(C^{*}_{r}(G_{e})).}$$\scriptstyle{\operatorname{id}-\widehat{\alpha}_{0}}$$\scriptstyle{\operatorname{id}-\widehat{\alpha}_{1}}$

###### Proof.

By Proposition 5.3, we have that the dual graph of $T$ is a row-finite,
finitely aligned multitree with no sources. The vertex stabilisers of
$\widehat{T}$ are the edge stabilisers of $T$, which are amenable because they
are subgroups of the vertex stabilisers of $T$, which are assumed to be
amenable. Moreover, the action of $G$ on $\partial\widehat{T}\cong\partial T$
is amenable by [7, Proposition 5.2.1 and Lemma 5.2.6]. So Remark 4.9 allows us
to apply Theorem 4.8 for the dual graph to get a six-term exact sequence

${\bigoplus_{[e]\in
G\backslash\widehat{T}^{0}}K_{0}(C^{*}_{r}(G_{e}))}$${\bigoplus_{[ef]\in
G\backslash\widehat{T}^{1}}K_{0}(C^{*}_{r}(G_{ef}))}$${K_{0}(C_{0}(\partial\widehat{T})\rtimes_{\tau_{\partial},r}G)}$${K_{1}(C_{0}(\partial\widehat{T})\rtimes_{\tau_{\partial},r}G)}$${\bigoplus_{[ef]\in
G\backslash\widehat{T}^{1}}K_{1}(C^{*}_{r}(G_{ef}))}$${\bigoplus_{[e]\in
G\backslash\widehat{T}^{0}}K_{1}(C^{*}_{r}(G_{e})),}$$\scriptstyle{\operatorname{id}-\widehat{\alpha}_{0}}$$\scriptstyle{\operatorname{id}-\widehat{\alpha}_{1}}$

where $\widehat{\alpha}_{i}=\sum_{[ef]\in
G\backslash\widehat{T}^{1}}(\theta_{ef})_{*,i}$. But $\widehat{T}^{0}=T^{1}$
and $\widehat{T}^{1}=T^{2}_{r}$ by definition, and Proposition 5.3 implies
that $C_{0}(\partial\widehat{T})\rtimes_{\tau_{\partial},r}G\cong
C_{0}(\partial T)\rtimes_{\tau_{\partial},r}G$, which together imply the
theorem. ∎

######  Remark 5.5.

The assumption in Theorem 5.4 that all vertex stabilisers are amenable means
that the reduced group $C^{*}$-algebras and crossed products are naturally
isomorphic to their respective full versions. Thus, Theorem 5.4 can be viewed
as a special case of [17, Corollary 6.4], which provides the analogous six-
term exact sequence for the full group $C^{*}$-algebras and crossed products,
without needing any amenability assumptions. However, the maps in the sequence
in [17] are described in terms of the $KK$-classes of $C^{*}$-correspondences
rather than in terms of induced homomorphisms. It is not immediately clear to
the authors how these pictures correspond to each other. However, we note that
in the special cases where all vertex stabilisers are trivial or where all
vertex stabilisers are infinite cyclic, the formulas for
$\operatorname{id}-\widehat{\alpha}_{i}$ in Theorem 5.4 reduce to the formulas
for $\Lambda_{i}$ in [17, Section 6].

######  Remark 5.6.

In [8], the authors associate to each locally finite nonsingular graph of
discrete groups $\mathcal{G}$ a universal $C^{*}$-algebra
$C^{*}(\mathcal{G})$, and show that $C^{*}(\mathcal{G})$ is stably isomorphic
to the full crossed product of the fundamental group of $\mathcal{G}$ acting
on the Bass–Serre tree of $\mathcal{G}$ [8, Theorem 4.1]. This means that in
the setting where all vertex stabilisers are amenable, Theorem 5.4 can be
rephrased in terms of the $K$-theory of the graph-of-groups $C^{*}$-algebra
$C^{*}(\mathcal{G})$, just as is done in [17, Theorem 6.2].

##  6\. Properties of the action $G\curvearrowright\partial E$

In this section we examine dynamical properties for the action of a discrete
group $G$ on the boundary $\partial E$ of a row-finite, finitely aligned
multitree $E$ with no sources, induced from an action of $G$ on $E$. We
investigate minimality in Section 6.1, local contractivity in Section 6.2,
topological freeness in Section 6.3, and finally amenability in Section 6.4.
Note that it follows from Proposition 5.3 that the results in this section
generalise the results in [8, Section 5], which deals only with group actions
on undirected trees. We will use the same notation as in Notation 4.18.

###  6.1. Minimality

Recall that an action of a discrete group $G$ on a locally compact Hausdorff
space $X$ is called minimal if every orbit of points of $X$ is dense in $X$.
We have the following characterisation in our context.

######  Proposition 6.1.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources. The induced action $G\curvearrowright\partial
E$ is minimal if and only if the graph $\Gamma$ is cofinal, that is, for every
infinite path $\lambda\in\Gamma^{\infty}$ and every vertex $v\in\Gamma^{0}$,
there is some vertex $w$ on $\lambda$ so that there is a path in $\Gamma$ from
$w$ to $v$.

###### Proof.

First suppose that the action $G\curvearrowright\partial E$ is minimal. Fix an
infinite path $\lambda$ in $\Gamma$ and a vertex $v\in\Gamma^{0}$. Then
$\lambda$ and $v$ lift to some infinite path $\lambda^{\prime}\in E^{\infty}$
and some vertex $v^{\prime}\in E^{0}$ respectively. By minimality, there is
$g\in G$ so that $g\cdot[\lambda^{\prime}]=[g\cdot\lambda^{\prime}]\in
Z_{\partial}(v^{\prime})$. This means there is an infinite path
$\mu^{\prime}\in v^{\prime}E^{\infty}$ so that $\mu^{\prime}$ and
$g\cdot\lambda^{\prime}$ agree after some vertex, say $w^{\prime}\in E^{0}$.
So there is a path from $w^{\prime}$ (which is on $g\cdot\lambda^{\prime}$) to
$v^{\prime}$, which descends to a path from $w\in\Gamma^{0}$ to $v$. But since
$w^{\prime}$ is in the same orbit as $g^{-1}\cdot w^{\prime}$, which is on
$\lambda^{\prime}$, we get that $w$ is on $\lambda$, so there is a path from
some $w$ on $\lambda$ to $v$.

Now suppose that $\Gamma$ is cofinal. Fix $\lambda^{\prime}\in E^{\infty}$ and
$v^{\prime}\in E^{0}$ arbitrary. It is enough to prove that there is $g\in G$
such that $g\cdot[\lambda^{\prime}]=[g\cdot\lambda^{\prime}]\in
Z_{\partial}(v^{\prime})$. Write $\lambda$ and $v$ for the images of
$\lambda^{\prime}$ and $v^{\prime}$ respectively under the quotient map. Then
$\lambda$ is an infinite path in $\Gamma$ and $v\in\Gamma^{0}$ is a vertex, so
there is some vertex $w$ on $\lambda$ such that there is a path in $\Gamma$
from $w$ to $v$. Now, there is a lift $w^{\prime}\in E^{0}$ of $w$ such that
$w^{\prime}$ is on $\lambda^{\prime}$. Also, the path from $w$ to $v$ lifts to
a path from some vertex $w^{\prime\prime}\in E^{0}$ to $v^{\prime}$, and
$w^{\prime\prime}$ is in the same vertex orbit as $w^{\prime}$. Let $g\in G$
be an element such that $g\cdot w^{\prime}=w^{\prime\prime}$. Then
$w^{\prime\prime}$ will be on $g\cdot\lambda^{\prime}$, so
$g\cdot\lambda^{\prime}$ is shift-tail equivalent to an infinite path with
range $v^{\prime}$. Hence $g\cdot[\lambda^{\prime}]\in
Z_{\partial}(v^{\prime})$. ∎

###  6.2. Local contractivity

Recall that an action of a discrete group $G$ on a locally compact Hausdorff
space $X$ is called locally contractive if for every non-empty open set
$U\subseteq X$ there is a non-empty open set $V\subseteq U$ and an element
$g\in G$ such that $g\overline{V}\subsetneq V$. Like in [8], we give a
sufficient condition for the action $G\curvearrowright\partial E$ to be
locally contractive.

First recall that a loop $\eta=\eta_{1}\cdots\eta_{m}$ in a directed graph
$\Gamma$ has an entrance at vertex $x\in\Gamma^{0}$ if there exists $\eta_{i}$
such that $r(\eta_{i})=x$ and $|r^{-1}(x)|\geq 2$.

######  Proposition 6.2.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources. Suppose that for every vertex $v\in\Gamma^{0}$
there is a loop $\eta=\eta_{1}\dots\eta_{m}$ in $\Gamma$ such that

  * (i)

$\eta$ has an entrance, or $[G_{r(\eta_{i})}:G_{\eta_{i}}]\geq 2$ for some
$1\leq i\leq m$; and

  * (ii)

there is a path from $r(\eta)$ to $v$.

Then the induced action $G\curvearrowright\partial E$ is locally contractive.

###### Proof.

Let $U\subseteq\partial E$ be a non-empty open subset, and let $v^{\prime}\in
E^{0}$ be such that $Z_{\partial}(v^{\prime})\subseteq U$. Let
$v\in\Gamma^{0}$ be the image of $v^{\prime}$ under the quotient map. Let
$\eta$ be as in the assumption. The path from $r(\eta)$ to $v$ lifts to a path
from some vertex $w^{\prime}\in E^{0}$ to $v^{\prime}$, and so
$Z_{\partial}(w^{\prime})\subseteq Z_{\partial}(v^{\prime})\subseteq U$ by
Lemma 4.5. Now $\eta$ lifts to some path $\eta^{\prime}$ from some vertex
$w^{\prime\prime}\in E^{0}$ to $w^{\prime}$, and $w^{\prime\prime}$ is in the
vertex orbit of $w^{\prime}$. So $w^{\prime\prime}=g\cdot w^{\prime}$ for some
$g\in G$, and $g\cdot Z_{\partial}(w^{\prime})=Z_{\partial}(g\cdot
w^{\prime})\subseteq Z_{\partial}(w^{\prime})$ by Lemma 4.5. Note that
$Z_{\partial}(w^{\prime})$ is closed, so all that is left to show is that the
containment $Z(g\cdot w^{\prime})\subseteq Z(w^{\prime})$ is proper.

By assumption, there is $x\in\Gamma^{0}$ such that $\eta$ either has an
entrance at $x$, or that there is $\eta_{i}$ with $r(\eta_{i})=x$ and
$[G_{x}:G_{\eta_{i}}]\geq 2$. Write $x^{\prime}\in E^{0}$ for a lift of $x$
which lies on $\eta^{\prime}$. Note that by the orbit-stabiliser theorem,
$\sum_{e\in x\Gamma^{1}}[G_{x}:G_{e}]$ is equal to the number of edges in
$E^{1}$ with range $x^{\prime}$; and the assumption ensures that this number
is at least $2$. In particular, if $e^{\prime}$ is the edge on $\eta^{\prime}$
with range $x^{\prime}$, then there is a distinct edge $e^{\prime\prime}\in
x^{\prime}E^{1}$. Note that $w^{\prime}\leq s(e^{\prime\prime})$, so by Lemma
4.5 we have that $Z_{\partial}(s(e^{\prime\prime}))\subseteq
Z_{\partial}(w^{\prime})$.

We claim that $Z_{\partial}(s(e^{\prime\prime}))\cap Z_{\partial}(g\cdot
w^{\prime})=\emptyset$. Indeed, suppose for a contradiction that
$Z_{\partial}(s(e^{\prime\prime}))\cap Z_{\partial}(g\cdot
w^{\prime})\neq\emptyset$. Then Proposition 4.4 implies that
$s(e^{\prime\prime})$ and $w^{\prime\prime}=g\cdot w^{\prime}$ have a common
upper bound $u^{\prime}\in E^{0}$. Write $\lambda_{1},\lambda_{2}$ for the
paths from $u^{\prime}$ to $s(e^{\prime\prime})$ and $w^{\prime\prime}$
respectively, and write $\mu$ for subpath of $\eta^{\prime}$ from
$w^{\prime\prime}$ to $x^{\prime}$. Then $e^{\prime\prime}\lambda_{1}$ and
$\mu\lambda_{2}$ are both paths from $u^{\prime}$ to $x^{\prime}$, and they
are distinct since they have distinct range-most edges by construction. This
contradicts that $E$ is a multitree, and so
$Z_{\partial}(s(e^{\prime\prime}))\cap Z_{\partial}(g\cdot
w^{\prime})=\emptyset$. This implies that the containment $Z_{\partial}(g\cdot
w^{\prime})\subseteq Z_{\partial}(w^{\prime})$ is proper, completing the
proof. ∎

###  6.3. Topological freeness

Recall that an action of a group $G$ on a locally compact Hausdorff space $X$
is called topologically free if for all $g\in G\setminus\\{1\\}$, the set
$\\{x\in X:g\cdot x\neq x\\}$ is dense in $X$. For $G$ discrete, an action
$G\curvearrowright X$ is topologically free if and only if the set of points
in $X$ with trivial isotropy is dense in $X$.

It seems to be difficult in general to characterise topological freeness of an
action $G\curvearrowright\partial E$ in terms of the corresponding quotient
directed graph and the stabiliser subgroups. However, we have
characterisations in the special cases where $G$ acts freely on $E$
(Proposition 6.4) and where all vertex stabilisers are infinite cyclic
(Proposition 6.6).

We start with a lemma.

######  Lemma 6.3.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources. For any finite path $\alpha=e_{1}e_{2}\dots
e_{n}$ in $\Gamma$ and any lift $v^{\prime}\in E^{0}$ of $r(\alpha)$, the
number of lifts of $\alpha$ with range $v^{\prime}$ is equal to

$[G_{r(e_{1})}:G_{e_{1}}][G_{r(e_{2})}:G_{e_{2}}]\cdots[G_{r(e_{n})}:G_{e_{n}}].$

###### Proof.

The claim follows from the fact that for any edge $e\in\Gamma^{1}$ and any
lift $v^{\prime}\in E^{0}$ of $r(e)$, the number of lifts of $e$ with range
$v^{\prime}$ is equal to $[G_{r(e)}:G_{e}]$. ∎

We start with the case where $G$ acts freely on $E$. First, recall that a
directed graph $\Gamma$ is called aperiodic if every loop has an entrance.

######  Proposition 6.4.

Suppose that a discrete group $G$ acts freely on a row-finite, finitely
aligned multitree $E$ with no sources. The induced action
$G\curvearrowright\partial E$ is topologically free if and only if $\Gamma$ is
aperiodic.

###### Proof.

First suppose that $\Gamma$ is aperiodic. This means that for every vertex
$v\in\Gamma^{0}$, there is an infinite path $\lambda\in v\Gamma^{\infty}$ with
aperiodic edge sequence [6, Lemma 3.2] (note the opposite convention for path
direction used in that paper). To show that $G\curvearrowright\partial E$ is
topologically free, we prove that every cylinder set in $\partial E$ contains
a point with trivial isotropy.

Fix $v^{\prime}\in E^{0}$. Write $v\in\Gamma^{0}$ for its image under the
quotient map, and let $\lambda\in v\Gamma^{\infty}$ be an infinite path with
aperiodic edge sequence. Let $\lambda^{\prime}\in v^{\prime}E^{\infty}$ be a
lift of $\lambda$. We claim that $[\lambda^{\prime}]\in
Z_{\partial}(v^{\prime})$ has trivial isotropy. Indeed, suppose that $g\in G$
fixes $[\lambda^{\prime}]$, so there exist $m,n\in\mathbb{N}$ such that
$\sigma^{m}(g\cdot\lambda^{\prime})=\sigma^{n}(\lambda^{\prime})$. This
implies that $\sigma^{m}(\lambda)$ and $\sigma^{n}(\lambda)$ are the same
infinite path in $\Gamma^{\infty}$, and so by aperiodicity of $\lambda$ we
must have that $m=n$. This means that $g$ fixes the infinite path
$\sigma^{n}(\lambda^{\prime})$ and hence the vertex
$r(\sigma^{n}(\lambda^{\prime}))$. But since $G$ is assumed to act freely on
$E$, this implies that $g$ is the identity element. Hence $[\lambda^{\prime}]$
has trivial isotropy as claimed.

Now suppose that $\Gamma$ is not aperiodic. Then $G$ is not trivial (since if
it were, then $\Gamma=G\backslash E$ would simply be the multitree $E$, which
is aperiodic since it cannot contain any loops), and there is a loop $\eta$ in
$\Gamma$ with no entrance. Let $v=r(\eta)$, and let $v^{\prime}\in E^{0}$ be a
lift of $v$. First we claim that there is a unique lift $\lambda^{\prime}\in
v^{\prime}E^{\infty}$ of $\eta^{\infty}:=\eta\eta\cdots$ with range
$v^{\prime}$. Indeed, since all vertex stabilisers are trivial, we have
$[G_{r(e)}:G_{e}]=1$ for all $e\in\Gamma^{1}$, and so Lemma 6.3 implies that
for every lift $v^{\prime\prime}\in E^{0}$ of $v$, there is a unique lift
$\eta^{\prime\prime}\in v^{\prime\prime}E^{*}$ of $\eta$. This in turn implies
that $\lambda^{\prime}$ is the only lift of $\eta^{\infty}$ with range
$v^{\prime}$, as claimed.

Now since $\eta$ has no entrance, the only infinite path in $\Gamma$ with
range $v$ is the infinite path $\eta^{\infty}$. This implies that
$\lambda^{\prime}$ is the only infinite path in $E$ with range $v^{\prime}$,
and so $Z_{\partial}(v^{\prime})$ is the singleton set
$\\{[\lambda^{\prime}]\\}$. We claim that $[\lambda^{\prime}]$ has non-trivial
isotropy. Write $m$ for the length of $\eta$. Then
$\sigma^{m}(\lambda^{\prime})$ is the unique lift of $\eta^{\infty}$ with
range $r(\sigma^{m}(\lambda^{\prime}))$. Since
$r(\sigma^{m}(\lambda^{\prime}))$ is also a lift of $v\in\Gamma^{0}$, it is in
the same orbit as $v^{\prime}$, and so there is some (necessarily non-
identity) $g\in G$ such that $g\cdot
v^{\prime}=r(\sigma^{m}(\lambda^{\prime}))$. But $g\cdot\lambda^{\prime}$ must
be a lift of $\eta^{\infty}$ with range $r(\sigma^{m}(\lambda^{\prime}))$, and
so uniqueness gives that
$g\cdot\lambda^{\prime}=\sigma^{m}(\lambda^{\prime})$. Hence
$g\cdot[\lambda^{\prime}]=[\lambda^{\prime}]$, showing that
$[\lambda^{\prime}]$ has non-trivial isotropy.

This implies that $Z_{\partial}(v^{\prime})$ is an open set in $\partial E$
which does not contain a point with trivial isotropy. Hence the set of points
in $\partial E$ with trivial isotropy is not dense in $\partial E$, so the
action $G\curvearrowright\partial E$ is not topologically free. ∎

We now turn to the situation where the vertex stabilisers of the action
$G\curvearrowright E$ are all infinite cyclic. Recall from Notation 4.18 the
definition of the integers $\omega_{e},\omega_{\overline{e}}$ for each
$e\in\Gamma^{1}$. We need the following concepts from [8], adapted to our
context.

######  Definition 6.5.

For a path $\alpha=e_{1}e_{2}\dots e_{n}$ in $\Gamma$, we define the signed
index ratio $q(\alpha)\in\mathbb{Q}^{\times}$ by

$q(\alpha)=\prod_{i=1}^{n}\frac{\omega_{\overline{e_{i}}}}{\omega_{e_{i}}}.$

Consistently with [8], for $q\in\mathbb{Q}$ we write $\langle q\rangle$ for
the smallest positive integer denominator that can be used to express $q$ as a
fraction. Recall that for any infinite path $\lambda=e_{1}e_{2}\cdots$ and any
$n\geq 1$, we denote by $\lambda_{n}$ the path $e_{1}\ldots e_{n}$. For any
finite path $\alpha=e_{1}e_{2}\dots e_{k}$ and any $1\leq n\leq k$, we
similarly denote by $\alpha_{n}$ the path $e_{1}\dots e_{n}$.

######  Proposition 6.6.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources, and suppose that $G_{v}\cong\mathbb{Z}$ for all
$v\in E^{0}$. The induced action $G\curvearrowright\partial E$ is
topologically free if and only if for all $v\in\Gamma^{0}$ there exists an
infinite path $\lambda=e_{1}e_{2}\ldots\in v\Gamma^{\infty}$ such that

$\limsup_{k\to\infty}\,\langle q(\lambda_{k-1})/\omega_{e_{k}}\rangle=\infty.$

######  Remark 6.7.

We note that the terms $\langle q(\gamma_{n})\rangle$ appearing in condition
(2) of [8, Theorem 7.5] are not quite correct, and should be replaced with
$\langle q(\gamma_{n})/\omega_{e_{n+1}}\rangle$ for respective $n\geq 1$.
However, this does not affect the validity of the other results in [8].

To prove Proposition 6.6, we need the following string of lemmas.

######  Lemma 6.8.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources, and suppose that $G_{v}\cong\mathbb{Z}$ for all
$v\in E^{0}$. Fix a finite path $\alpha=e_{1}e_{2}\dots e_{n}$ in $\Gamma$.
For any lift $\alpha^{\prime}\in E^{n}$ of $\alpha$, an element
$m1_{r(\alpha^{\prime})}\in G_{r(\alpha^{\prime})}$, $m\in\mathbb{Z}$ fixes
$\alpha^{\prime}$ if and only if $\langle
q(\alpha_{k-1})/\omega_{e_{k}}\rangle\mid m$ for all $1\leq k\leq n$.

###### Proof.

We show by induction on $n$ that $m1_{r(\alpha^{\prime})}$ fixes
$\alpha^{\prime}$ if and only if $mq(\alpha_{k-1})/\omega_{e_{k}}$ is an
integer for all $1\leq k\leq n$; the claim of the lemma follows from this.
(Note that $\alpha_{0}:=r(\alpha)$.) Moreover, we show in the same induction
that if $m1_{r(\alpha^{\prime})}$ fixes $\alpha^{\prime}$, then $mq(\alpha)$
is an integer and $m1_{r(\alpha^{\prime})}=mq(\alpha)1_{s(\alpha^{\prime})}$
up to sign.

If $n=1$, then $\alpha=e_{1}$, and for any lift $e^{\prime}$ of $e_{1}$, we
have that $G_{e^{\prime}}$ is the subgroup of $G_{r(e^{\prime})}$ of index
$\lvert\omega_{e_{1}}\rvert$. Hence $m1_{r(\alpha^{\prime})}$ fixes
$\alpha^{\prime}$ if and only if $\omega_{e_{1}}\mid m$, or equivalently, if
$m/\omega_{e_{1}}$ is an integer. Since $q(\alpha_{0})=q(r(\alpha))=1$ the
first claim holds for $n=1$. For the second claim, note that
$m1_{r(e^{\prime})}=(m/\omega_{e_{1}})1_{e^{\prime}}=(m\omega_{\overline{e_{1}}}/\omega_{e_{1}})1_{s(e^{\prime})}$
up to sign, and $m\omega_{\overline{e_{1}}}/\omega_{e_{1}}=mq(e_{1})$ is an
integer since $m/\omega_{e_{1}}$ is.

Now suppose that the inductive claim holds for all paths of length $n=l$. Fix
a finite path $\alpha=e_{1}e_{2}\dots e_{l+1}$ in $\Gamma$ of length $l+1$,
and a lift $\alpha^{\prime}=e^{\prime}_{1}e^{\prime}_{2}\dots
e^{\prime}_{l+1}\in E^{l+1}$ of $\alpha$. Fix $m\in\mathbb{Z}$. Now
$m1_{r(\alpha^{\prime})}$ fixes $\alpha^{\prime}$ if and only if it fixes both
$\alpha^{\prime}_{l}$ and $e^{\prime}_{l+1}$. The inductive assumption gives
that $m1_{r(\alpha^{\prime})}$ fixes $\alpha^{\prime}_{l}$ if and only if
$mq(\alpha_{k-1})/\omega_{e_{k}}$ is an integer for all $1\leq k\leq l$, and
that in this case, $m^{\prime}:=mq(\alpha_{l})$ is an integer and
$m^{\prime}1_{s(\alpha^{\prime}_{l})}=m1_{r(\alpha^{\prime})}$ (up to sign) in
$G$. Now from the $n=1$ case, we know that
$m^{\prime}1_{s(\alpha^{\prime}_{l})}=m^{\prime}1_{r(e^{\prime}_{l+1})}$ fixes
$e^{\prime}_{l+1}$ if and only if
$m^{\prime}/\omega_{e_{l+1}}=mq(\alpha_{l})/\omega_{e_{l+1}}$ is an integer.
This proves the first inductive claim for $n=l+1$.

For the second claim, note that the inductive assumption gives that
$mq(\alpha_{l})$ is an integer and that
$m1_{r(\alpha^{\prime})}=mq(\alpha_{l})1_{s(\alpha^{\prime}_{l})}=mq(\alpha_{l})1_{r(e^{\prime}_{l+1})}$
up to sign. But the $n=1$ case for the edge $e^{\prime}_{l+1}$ (with
$m^{\prime}$ in place of $m$) gives that
$m^{\prime}q(e_{l+1})=mq(\alpha_{l})q(e_{l+1})=mq(\alpha)$ is an integer and
that
$mq(\alpha_{l})1_{r(e^{\prime}_{l+1})}=mq(\alpha)1_{s(e^{\prime}_{l+1})}=mq(\alpha)1_{s(\alpha^{\prime})}$
up to sign. Thus the second claim also holds.

Hence the inductive claim holds for all $n\geq 1$, and we are done. ∎

######  Corollary 6.9.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources, and suppose that $G_{v}\cong\mathbb{Z}$ for all
$v\in E^{0}$. Every finite path in $E$ has non-trivial isotropy.

######  Lemma 6.10.

Suppose that a discrete group $G$ acts on a row-finite, finitely aligned
multitree $E$ with no sources, and suppose that $G_{v}\cong\mathbb{Z}$ for all
$v\in E^{0}$. Fix an infinite path
$\lambda=e_{1}e_{2}\ldots\in\Gamma^{\infty}$ and a lift $\lambda^{\prime}\in
E^{\infty}$ of $\lambda$. We have

$G_{\lambda^{\prime}}=0\quad\iff\quad\limsup_{k\to\infty}\,\langle
q(\lambda_{k-1})/\omega_{e_{k}}\rangle=\infty.$

###### Proof.

First note that $G_{\lambda^{\prime}}$ is a subgroup of
$G_{r(\lambda^{\prime})}$. Now, an element $m1_{r(\lambda^{\prime})}$,
$m\in\mathbb{Z}$ fixes $\lambda^{\prime}$ if and only if it fixes
$\lambda^{\prime}_{k}$ for all $k\geq 1$. By Lemma 6.8 this is equivalent to
the condition that $\langle q(\lambda_{k-1})/\omega_{e_{k}}\rangle\mid m$ for
all $k\geq 1$. If $\limsup_{k\to\infty}\,\langle
q(\lambda_{k-1})/\omega_{e_{k}}\rangle=\infty$, then the set $\\{\langle
q(\lambda_{k-1})/\omega_{e_{k}}\rangle:k\geq 1\\}$ is unbounded, and so the
only value of $m$ satisfying $\langle
q(\lambda_{k-1})/\omega_{e_{k}}\rangle\mid m$ for all $k\geq 1$ is $m=0$.
Hence $G_{\lambda^{\prime}}=0$ in this case. On the other hand, if
$\limsup_{k\to\infty}\,\langle q(\lambda_{k-1})/\omega_{e_{k}}\rangle<\infty$,
then the set $\\{\langle q(\lambda_{k-1})/\omega_{e_{k}}\rangle:k\geq 1\\}$ is
bounded. This set therefore has a (finite) least common multiple $M$, and by
definition we have that $\langle q(\lambda_{k-1})/\omega_{e_{k}}\rangle\mid M$
for all $k\geq 1$. Hence $M1_{r(\lambda^{\prime})}\neq 0$ fixes
$\lambda^{\prime}$, so $G_{\lambda^{\prime}}$ is not trivial in this case. ∎

###### Proof of Proposition 6.6.

For the forward direction, suppose that the action $G\curvearrowright\partial
E$ is topologically free, and fix $v\in\Gamma^{0}$. Choose any lift
$v^{\prime}\in E^{0}$ of $v$. Since $G\curvearrowright\partial E$ is
topologically free, there is some infinite path $\lambda^{\prime}\in
v^{\prime}E^{\infty}$ such that $[\lambda^{\prime}]$ has trivial isotropy. In
particular, $G_{\lambda^{\prime}}=0$, and so writing $\lambda\in
v\Gamma^{\infty}$ for the image of $\lambda^{\prime}$ under the quotient map,
Lemma 6.10 shows that $\lambda$ satisfies the required condition.

We now prove the backward direction. Fix $v^{\prime}\in E^{0}$ and $g\in
G\setminus\\{1\\}$; it is enough to show that there is some infinite path
$\lambda^{\prime}\in v^{\prime}E^{\infty}$ such that $g$ does not fix
$[\lambda^{\prime}]$. Write $v\in\Gamma^{0}$ for the image of $v^{\prime}$
under the quotient map. By assumption, there is an infinite path
$\lambda=e_{1}e_{2}\ldots\in v\Gamma^{\infty}$ satisfying
$\limsup_{k\to\infty}\,\langle q(\lambda_{k-1})/\omega_{e_{k}}\rangle=\infty$.
If $g$ does not fix $[\lambda^{\prime}]$ for any lift $\lambda^{\prime}\in
v^{\prime}E^{\infty}$ of $\lambda$, then we are done, so suppose that there is
some lift $\lambda^{\prime\prime}\in v^{\prime}E^{\infty}$ of $\lambda$ such
that $g\cdot[\lambda^{\prime\prime}]=[\lambda^{\prime\prime}]$. This means
that there are $m,n\in\mathbb{N}$ such that
$\sigma^{m}(g\cdot\lambda^{\prime\prime})=\sigma^{n}(\lambda^{\prime\prime})$.

First we claim that $m\neq n$. Suppose for a contradiction that $m=n$. Then
$g$ fixes $\sigma^{n}(\lambda^{\prime\prime})$, so
$\sigma^{n}(\lambda^{\prime\prime})$ has non-trivial isotropy. This means that
the stabiliser $G_{\sigma^{n}(\lambda^{\prime\prime})}$ is some finite-index
subgroup of $G_{r(\sigma^{n}(\lambda^{\prime\prime}))}$. Now by Corollary 6.9,
we have that $\lambda^{\prime\prime}_{n}$ also has non-trivial isotropy, and
thus $G_{\lambda^{\prime\prime}_{n}}$ is also some finite-index subgroup of
$G_{r(\sigma^{n}(\lambda^{\prime\prime}))}=G_{s(\lambda^{\prime\prime}_{n})}$.
Since $G_{r(\sigma^{n}(\lambda^{\prime\prime}))}\cong\mathbb{Z}$, the
intersection of two finite-index subgroups is again a finite-index subgroup,
and so is in particular not the trivial subgroup $\\{0\\}$. This means that
$G_{\sigma^{n}(\lambda^{\prime\prime})}\cap G_{\lambda^{\prime\prime}_{n}}$ is
non-trivial. But an element in $G_{r(\sigma^{n}(\lambda^{\prime\prime}))}$
fixes $\lambda^{\prime\prime}$ if and only if it fixes both
$\lambda^{\prime\prime}_{n}$ and $\sigma^{n}(\lambda^{\prime\prime})$, so
$G_{\lambda^{\prime\prime}}=G_{\sigma^{n}(\lambda^{\prime\prime})}\cap
G_{\lambda^{\prime\prime}_{n}}$. So $G_{\lambda^{\prime\prime}}$ is non-
trivial, contradicting Lemma 6.10 and the choice of $\lambda^{\prime\prime}$.
Hence $m\neq n$.

Now suppose that $m<n$. Writing
$\mu^{\prime\prime}=\sigma^{n}(\lambda^{\prime\prime})$, we have that
$g\cdot\mu^{\prime\prime}=\sigma^{n-m}(\mu^{\prime\prime})$. This implies that
$\mu:=\sigma^{n}(\lambda)$, which is the image of $\mu^{\prime\prime}$ under
the quotient map, has the form $\eta\eta\cdots$ for some cycle $\eta$ in
$\Gamma$ of length $n-m$. We claim that $\langle q(\eta)\rangle>1$. Suppose
for a contradiction that $\langle q(\eta)\rangle=1$, so $q(\eta)$ is an
integer. Write $a_{k}:=q(\lambda_{k-1})/\omega_{e_{k}}$ for $k\geq 1$. Since
$\sigma^{n}(\lambda)$ is periodic (with period $n-m$), we get that
$a_{k+n-m}=q(\eta)a_{k}$ for all $k>n$. Since $q(\eta)$ is an integer, this
implies that $\langle a_{k+n-m}\rangle\leq\langle a_{k}\rangle$ for all $k>n$,
and so the sequence $\\{\langle a_{k}\rangle\\}_{k=1}^{\infty}=\\{\langle
q(\lambda_{k-1})/\omega_{e_{k}}\rangle\\}_{k=1}^{\infty}$ is bounded,
contradicting the choice of $\lambda$. Hence $\langle q(\eta)\rangle>1$.

Now we must have that $\lvert\omega_{e}\rvert>1$ for some edge $e$ in $\eta$;
otherwise $q(\eta)$ would be an integer and hence $\langle q(\eta)\rangle=1$,
contradicting the above. By Lemma 6.3, this means that for every lift
$w^{\prime}\in E^{0}$ of $r(\eta)$, there is more than one lift of $\eta$ with
range $w^{\prime}$.

We are now ready to construct an infinite path $\lambda^{\prime}\in
v^{\prime}E^{\infty}$ such that
$g\cdot[\lambda^{\prime}]\neq[\lambda^{\prime}]$. Notice that
$\mu^{\prime\prime}_{n-m}$ is a lift of $\eta$ with range
$r(\mu^{\prime\prime})$. By the above discussion, there is a distinct lift
$\eta^{\prime}$ of $\eta$ with the same range. Let $\xi^{\prime}$ be a lift of
$\eta\eta\cdots$ with range $s(\eta^{\prime})$, and write
$\lambda^{\prime}=\lambda^{\prime\prime}_{n}\eta^{\prime}\xi^{\prime}$. We
show that $g$ does not fix $[\eta^{\prime}\xi^{\prime}]=[\lambda^{\prime}]$.
Note that $g$ sends $r(\eta^{\prime})=r(\mu^{\prime\prime}_{n-m})$ to
$s(\mu^{\prime\prime}_{n-m})$, since
$g\cdot\mu^{\prime\prime}=\sigma^{n-m}(\mu^{\prime\prime})$. So
$g\cdot\eta^{\prime}\xi^{\prime}\in s(\mu^{\prime\prime}_{n-m})E^{\infty}$.

We claim that $Z_{\partial}(s(\eta^{\prime}))\cap
Z_{\partial}(s(\mu^{\prime\prime}_{n-m}))=\emptyset$; this would then imply
that $\xi^{\prime}\in s(\eta^{\prime})E^{\infty}$ (and therefore
$\lambda^{\prime}$) is not shift-tail equivalent to
$g\cdot\eta^{\prime}\xi^{\prime}\in s(\mu^{\prime\prime}_{n-m})E^{\infty}$
(and hence to $g\cdot\lambda^{\prime}$). Suppose for a contradiction that
$Z_{\partial}(s(\eta^{\prime}))\cap
Z_{\partial}(s(\mu^{\prime\prime}_{n-m}))\neq\emptyset$. Then Proposition 4.4
implies that $s(\eta^{\prime})$ and $s(\mu^{\prime\prime}_{n-m})$ have some
(minimal) common upper bound $u\in E^{0}$. By definition, this means that
there are paths $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ from $u$ to
$s(\eta^{\prime})$ and $s(\mu^{\prime\prime}_{n-m})$ respectively. But then
$\eta^{\prime}\alpha^{\prime}$ and
$\mu^{\prime\prime}_{n-m}\alpha^{\prime\prime}$ would both be paths from $u$
to $r(\mu^{\prime\prime})=r(\eta^{\prime})$, and they are distinct since
$\eta^{\prime}\neq\mu^{\prime\prime}_{n-m}$ by construction and since
$\eta^{\prime}$ and $\mu^{\prime\prime}_{n-m}$ have the same length. This
contradicts the fact that $E$ is a multitree, and hence
$Z_{\partial}(s(\eta^{\prime}))\cap
Z_{\partial}(s(\mu^{\prime\prime}_{n-m}))=\emptyset$ as claimed. So
$\lambda^{\prime}$ and $g\cdot\lambda^{\prime}$ are not shift-tail equivalent,
meaning that $g$ does not fix $[\lambda^{\prime}]\in
Z_{\partial}(v^{\prime})$, as required.

Finally suppose that $m>n$. Note that

$\sigma^{n}(g^{-1}\cdot\lambda^{\prime\prime})=g^{-1}\cdot\sigma^{n}(\lambda^{\prime\prime})=g^{-1}\cdot\sigma^{m}(g\cdot\lambda^{\prime\prime})=g^{-1}g\cdot\sigma^{m}(\lambda^{\prime\prime})=\sigma^{m}(\lambda^{\prime\prime}).$

Since $n<m$, the argument for the $m<n$ case (with $m$ and $n$ swapped) gives
the existence of an infinite path $\lambda^{\prime}\in v^{\prime}E^{\infty}$
such that $g^{-1}\cdot[\lambda^{\prime}]\neq[\lambda^{\prime}]$. But acting by
$g$ on both sides of the equation gives that $[\lambda^{\prime}]\neq
g\cdot[\lambda^{\prime}]$, which is what we needed. ∎

###  6.4. Amenability

For actions on undirected trees, we have that the amenability of all vertex
stabilisers ensures the amenability of the action $G\curvearrowright\partial
T$ [8, Theorem 5.29]. We would like something similar in the multitree
context. It is unclear to the authors what the situation is in general,
however we are able to prove Proposition 6.11, which is the corresponding
result for actions on directed trees.

In this section we use the formulation of an amenable group action given in
[4, Definition 2.1], which is as follows. For any locally compact group $G$,
write $\operatorname{Prob}(G)$ for the space of probability measures on $G$
equipped with the weak*-topology. Given $s,t\in G$, $f\in C_{c}(G)$ and
$m\in\operatorname{Prob}(G)$, set

$(s\cdot f)(t)=f(s^{-1}t)\quad\text{and}\quad(s\cdot m)(f)=m(s^{-1}\cdot f).$

An action of $G$ on some locally compact Hausdorff space $X$ is amenable if
there is a net $(m_{i})_{i\in I}$ of continuous maps $x\mapsto m_{i}^{x}$ from
$X$ to $\operatorname{Prob}(G)$ such that

$\lim_{i}\lVert s\cdot m_{i}^{x}-m_{i}^{s\cdot x}\rVert_{1}=0$

uniformly for $(x,s)$ in compact subsets of $X\times G$.

######  Proposition 6.11.

Let $G$ be a discrete group acting on a directed tree $T_{+}$. If for all
$v\in T^{0}$ the stabiliser $G_{v}\leq G$ is an amenable group, then the
induced action $G\curvearrowright\partial T_{+}$ is amenable.

We first prove the following lemma.

######  Lemma 6.12.

Let $G$ be a locally compact group, and let $X$ and $Y$ be locally compact
spaces on which $G$ acts. Suppose that $G\curvearrowright X$ is amenable, and
that there exists a continuous, $G$-equivariant map $\varphi\colon Y\to X$.
Then the action $G\curvearrowright Y$ is amenable.

###### Proof.

By definition of the amenability of $G\curvearrowright X$, there is a net
$(m_{i})_{i\in I}$ of continuous maps $x\mapsto m_{i}^{x}$ from $X$ to
$\operatorname{Prob}(G)$ such that

(6.1) $\lim_{i}\lVert s\cdot m_{i}^{x}-m_{i}^{s\cdot x}\rVert_{1}=0$

uniformly for $(x,s)$ in compact subsets of $X\times G$. Now since
$\varphi\colon Y\to X$ is assumed to be continuous, each $m_{i}\circ\varphi$
is a continuous map from $Y$ to $\operatorname{Prob}(G)$, where $y\mapsto
m_{i}^{\varphi(y)}$. Now, using Equation (6.1) and the $G$-equivariance of
$\varphi$, we get that

$\lim_{i}\lVert s\cdot(m_{i}\circ\varphi)^{y}-(m_{i}\circ\varphi)^{s\cdot
y}\rVert_{1}=\lim_{i}\lVert s\cdot m_{i}^{\varphi(y)}-m_{i}^{\varphi(s\cdot
y)}\rVert_{1}=\lim_{i}\lVert s\cdot
m_{i}^{\varphi(y)}-m_{i}^{s\cdot\varphi(y)}\rVert_{1}=0$

uniformly for $(\varphi(y),s)$ in compact subsets of $X\times G$. But
$\varphi\times\operatorname{id}_{G}$ maps any compact set in $Y\times G$ to a
compact subset of $X\times G$, since continuous images of compact sets are
compact and $\varphi\times\operatorname{id}_{G}$ is continuous. Hence the
limit is uniform for $(y,s)$ in compact subsets of $Y\times G$, and so
$G\curvearrowright Y$ is amenable. ∎

###### Proof of Proposition 6.11.

Write $T$ for the underlying undirected tree of $T_{+}$. The undirected tree
$T$ inherits an action of $G$ from $T_{+}$, with the same vertex stabilisers.
In particular, [8, Theorem 5.29] implies that the action
$G\curvearrowright\partial T$ is amenable. We aim to apply Lemma 6.12 with
$X=\partial T$ and $Y=\partial T_{+}$.

For $\mu\in T_{+}^{\infty}$ we write $[\mu]_{+}$ for the shift-tail
equivalence class of $\mu$ in $\partial T_{+}$ to distinguish it from the
shift-tail equivalence class $[\mu]$ in $\partial T$. Define a map
$\varphi\colon\partial T_{+}\to\partial T$ by $\varphi([\mu]_{+})=[\mu]$ for
all $\mu\in T_{+}^{\infty}$. This map is well-defined since for any
$\mu_{1},\mu_{2}\in T_{+}^{\infty}$, we have
$[\mu_{1}]_{+}=[\mu_{2}]_{+}\implies[\mu_{1}]=[\mu_{2}]$. Moreover, $\varphi$
is $G$-equivariant since for all $g\in G$ and $\mu\in T_{+}^{\infty}$ we have

$\varphi(g\cdot[\mu]_{+})=\varphi([g\cdot\mu]_{+})=[g\cdot\mu]=g\cdot[\mu]=g\cdot\varphi([\mu]_{+}).$

We now show that $\varphi$ is continuous. Recall that $\partial T$ has a basis
of (compact) open sets $\\{Z_{\partial}(e):e\in T^{1}\\}$, so it is enough to
show that $\varphi^{-1}(Z_{\partial}(e))$ is open in $\partial T_{+}$ for all
$e\in T^{1}$. Fix $e\in T^{1}$. We have

$\displaystyle\varphi^{-1}(Z_{\partial}(e))$
$\displaystyle=\\{[\mu]_{+}:\mu\in T_{+}^{\infty},\ [\mu]\in
Z_{\partial}(e)\\}$ $\displaystyle=\\{[\mu]_{+}:\mu\in T_{+}^{\infty},\
\exists\lambda\in s(e)T^{\infty}_{r}\ \text{such that}\ [\mu]=[\lambda]\
\text{and}\ \lambda_{1}\neq\overline{e}.\\}$

Now recall that $[\mu]=[\lambda]$ means that there are some $m,n\in\mathbb{N}$
and edges $e_{i}\in T^{1}$ for $1\leq i\leq n$ such that
$\lambda=e_{1}e_{2}\dots e_{n}\sigma^{m}(\mu)$. Note that we can write
$\sigma^{m}(\mu)=e_{n+1}\sigma^{m+1}(\mu)$ where $e_{n+1}\in T_{+}^{1}$, and
so $\lambda$ has the form

$\lambda=e_{1}e_{2}\dots e_{n}e_{n+1}\sigma^{m+1}(\mu).$

In particular, relabelling $n+1$ as $n$, we can assume without loss of
generality that $n\geq 1$ and that $e_{n}\in T_{+}^{1}$. Moreover, since
$\lambda$ is a reduced infinite path, we get that $\alpha:=e_{1}e_{2}\dots
e_{n}$ is also reduced. Note also that $\mu^{\prime}:=\sigma^{m+1}(\mu)$ is an
infinite directed path. So we can write

$\displaystyle\varphi^{-1}(Z_{\partial}(e))$
$\displaystyle\qquad=\\{[\mu^{\prime}]_{+}:\mu^{\prime}\in T_{+}^{\infty},\
\exists\alpha=e_{1}e_{2}\dots e_{n}\in s(e)T^{*}_{r}\ \text{such that}$
$\displaystyle\hskip 150.00023ptn\geq 1,\ e_{1}\neq\overline{e},\ e_{n}\in
T_{+}^{1}\ \text{and}\ \mu^{\prime}\in s(\alpha)T_{+}^{\infty}\\}$
$\displaystyle\qquad=\bigcup_{\begin{subarray}{c}\alpha=e_{1}\dots e_{n}\in
s(e)T^{*}_{r},\\\ n\geq 1,\ e_{1}\neq\overline{e},\ e_{n}\in
T_{+}^{1}\end{subarray}}Z_{\partial}(s(\alpha)),$

which implies that $\varphi^{-1}(Z_{\partial}(e))$ is open. Hence $\varphi$ is
continuous, so we can apply Lemma 6.12 to conclude that the action
$G\curvearrowright\partial T_{+}$ is amenable. ∎

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