#  Higman-Thompson groups from self-similar groupoid actions

Valentin Deaconu  Valentin Deaconu
Department of Mathematics (084)
University of Nevada
Reno NV 89557-0084
USA<EMAIL_ADDRESS>
###### Abstract.

Given a self-similar groupoid action $(G,E)$ on a finite directed graph, we
prove some properties of the corresponding ample groupoid of germs
$\mathcal{G}(G,E)$. We study the analogue of the Higman-Thompson group
associated to $(G,E)$ using $G$-tables and relate it to the topological full
group of $\mathcal{G}(G,E)$, which is isomorphic to a subgroup of unitaries in
the algebra $C^{*}(G,E)$. After recalling some concepts in groupoid homology,
we discuss the Matui’s AH-conjecture for $\mathcal{G}(G,E)$ in some particular
cases.

###### Key words and phrases:

Higman-Thompson group; Self-similar action; groupoid homology;
$C^{*}$-correspondence; Cuntz-Pimsner algebra.

###### 1991 Mathematics Subject Classification:

Primary 46L05.

##  1\. introduction

Self-similar group actions are defined using subgroups of the automorphism
group of a rooted tree which is viewed as the path space of a graph with one
vertex and $n$ edges. Given a finite directed graph $E$ with no sources, the
corresponding path space gives rise to a union of trees (or forest) $T_{E}$,
and it is natural to consider self-similar actions of subgroupoids $G$ of
PIso$(T_{E})$, the set of partial isomorphsims of $T_{E}$.

Given such a self-similar action $(G,E)$, we study the properties of the
generalized Exel-Pardo ample groupoid $\mathcal{G}(G,E)$. Since the source map
of the graph is in general not $G$-equivariant, this determines certain
modifications in the proofs of similar results from [6], where $G$ is a group
acting on $E$ by graph automorphisms. We study the topological full group
$\ldbrack\mathcal{G}(G,E)\rdbrack$ and we relate it to the Higman-Thompson
type group defined by $G$-tables for the action $(G,E)$. After recalling some
results about groupoid homology, we discuss the Matui’s AH-conjecture for
$\mathcal{G}(G,E)$ and related issues.

We begin by recalling the definition of a self-similar groupoid action $(G,E)$
from [8] and give several examples, illustrating some differences with the
case when $G$ is a group acting by automorphisms of the graph $E$. We recall
some facts about the structure of the associated $C^{*}$-algebra $C^{*}(G,E)$
defined using a $C^{*}$-correspondence over $C^{*}(G)$ and show that it
contains a copy of the graph algebra $C^{*}(E)$. We review the definition of
the ample groupoid of germs $\mathcal{G}(G,E)$, inspired from the Exel-Pardo
groupoid from [6] and describe when this groupoid is minimal and effective.

We then define the Higman-Thompson group $V_{E}(G)$ associated to $(G,E)$
using $G$-tables and describe a faithful unitary representation of $V_{E}(G)$
in the $C^{*}$-algebra $C^{*}(G,E)$. Topological full groups associated to
dynamical systems and to étale groupoids are complete invariants for
continuous orbit equivalence and groupoid isomorphism. They provide means of
constructing new countable groups with interesting properties.

After recalling the definion of homology of étale groupoids introduced by
Crainic and Moerdijk in [2] and Matui’s AH-conjecture, we specialize to the
groupoid $\mathcal{G}(G,E)$ associated to a self-similar action $(G,E)$ and
its topological full group $\ldbrack\mathcal{G}(G,E)\rdbrack$ which is
isomorphic to $V_{E}(G)$. The kernel $\mathcal{H}(G,E)$ of the groupoid
cocycle $\rho:\mathcal{G}(G,E)\to\mathbb{Z}$ plays an important rôle in the
computation of the groupoid homology $H_{i}(\mathcal{G}(G,E))$, particularly
when $G$ is transitive. We prove that in some cases, the index map
$I_{\mathcal{H}}:\ldbrack\mathcal{H}(G,E)\rdbrack\to H_{1}(\mathcal{H}(G,E))$
is surjective. We also show that in a particular example, the kernel of the
index map $\ldbrack\mathcal{G}(G,E)\rdbrack\to H_{1}(\mathcal{G}(G,E))$ is not
generated by transpositions.

##  2\. Self-similar actions of groupoids on graphs

Let $E=(E^{0},E^{1},r,s)$ be a finite directed graph with no sources. For
$k\geq 2$, define the set of paths of length $k$ in $E$ as

$E^{k}=\\{e_{1}e_{2}\cdots e_{k}:e_{i}\in E^{1},\;r(e_{i+1})=s(e_{i})\\}.$

The maps $r,s$ are naturally extended to $E^{k}$ by taking

$r(e_{1}e_{2}\cdots e_{k})=r(e_{1}),\;\;s(e_{1}e_{2}\cdots e_{k})=s(e_{k}).$

We denote by $E^{*}:=\bigcup_{k\geq 0}E^{k}$ the space of finite paths
(including vertices) and by $E^{\infty}$ the infinite path space of $E$ with
the usual topology given by the cylinder sets $Z(\alpha)=\\{\alpha\xi:\xi\in
E^{\infty}\\}$ for $\alpha\in E^{*}$. Note that $E^{\infty}$ is a Cantor space
precisely when $E$ satisfies condition (L).

We can visualize the set $E^{*}$ as indexing the vertices of a union of rooted
trees or forest $T_{E}$ given by $T_{E}^{0}=E^{*}$ and with edges

$T_{E}^{1}=\\{(\mu,\mu e):\mu\in E^{*},e\in
E^{1}\;\text{and}\;s(\mu)=r(e)\\}.$

######  Example 2.1.

For the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle
e_{2}$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{6}$$\scriptstyle e_{1}$

the forest $T_{E}$ looks like

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle
e_{1}$$\scriptstyle e_{3}$$\scriptstyle e_{1}e_{1}$$\vdots$$\scriptstyle
e_{1}e_{3}$$\vdots$$\scriptstyle e_{3}e_{2}$$\vdots$$\scriptstyle
e_{3}e_{6}$$\vdots$$\scriptstyle e_{2}$$\scriptstyle e_{6}$$\scriptstyle
e_{2}e_{1}$$\vdots$$\scriptstyle e_{2}e_{3}$$\vdots$$\scriptstyle
e_{6}e_{4}$$\vdots$$\scriptstyle e_{6}e_{5}$$\vdots$$\scriptstyle
e_{4}$$\scriptstyle e_{5}$$\scriptstyle e_{4}e_{2}$$\vdots$$\scriptstyle
e_{4}e_{6}$$\vdots$$\scriptstyle e_{5}e_{2}$$\vdots$$\scriptstyle
e_{5}e_{6}$$\vdots$

Recall that a partial isomorphism of the forest $T_{E}$ corresponding to a
given directed graph $E$ consists of a pair $(v,w)\in E^{0}\times E^{0}$ and a
bijection $g:vE^{*}\to wE^{*}$ such that

  * •

$g|_{vE^{k}}:vE^{k}\to wE^{k}$ is bijective for all $k\geq 1$.

  * •

$g(\mu e)\in g(\mu)E^{1}$ for $\mu\in vE^{*}$ and $e\in E^{1}$ with
$r(e)=s(\mu)$.

Here $vE^{k}$ denotes the set of paths $\mu\in E^{k}$ with $r(\mu)=v$ and
similarly for $vE^{*}$. The set of partial isomorphisms of $T_{E}$ forms a
discrete groupoid PIso$(T_{E})$ with unit space $E^{0}$. The identity
morphisms are $id_{v}:vE^{*}\to vE^{*}$, the inverse of $g:vE^{*}\to wE^{*}$
is $g^{-1}:wE^{*}\to vE^{*}$, and the multiplication is composition. We often
identify $v\in E^{0}$ with $id_{v}\in$ PIso$(T_{E})$.

In Example 2.1, the sum of the entries in each row of the graph adjacency
matrix $\displaystyle A_{E}=\left(\begin{array}[]{ccc}1&1&0\\\ 1&0&1\\\
0&2&0\end{array}\right)$ is the same, so the set of edges ending at each
vertex has the same cardinality, but in general it could happen that there is
no bijection $g:vE^{*}\to wE^{*}$ as above for $v\neq w$, see Example 2.8. In
that case, PIso$(T_{E})$ is a group bundle. If $E$ has a single vertex, then
PIso$(T_{E})=$Aut$(T_{E})$ is a group.

We are interested in self-similar actions of groupoids on the path space of
directed graphs $E$ as introduced and studied in [8].

A groupoid $G$ is a small category with inverses. We will use $d$ and $t$ for
the domain and target maps $d,t:G\to G^{(0)}$ to distinguish them from the
source and range maps $s,r:E^{1}\to E^{0}$ on directed graphs. For $u,v\in
G^{(0)}$, we write

$G_{u}=\\{g\in G:d(g)=u\\},\;\;G^{v}=\\{g\in
G:t(g)=v\\},\;\;G_{u}^{v}=G_{u}\cap G^{v}.$

The set of composable pairs is denoted $G^{(2)}$.

An étale groupoid is a topological groupoid where the target map $t$ (and
necessarily the domain map $d$) is a local homeomorphism (as a map from $G$ to
$G$). The unit space $G^{(0)}$ of an étale groupoid is always an open subset
of $G$.

######  Definition 2.2.

Let $G$ be an étale groupoid. A bisection is an open subset $U\subseteq G$
such that $d$ and $t$ are both injective when restricted to $U$.

Two units $x,y\in G^{(0)}$ belong to the same $G$-orbit if there exists $g\in
G$ such that $d(g)=x$ and $t(g)=y$. We denote by $G(x)$ the $G$-orbit of $x$.
When every $G$-orbit is dense in $G^{(0)}$, the groupoid $G$ is called
minimal.

The isotropy group of a unit $x\in G^{(0)}$ is the group

$G_{x}^{x}:=\\{g\in G\;|\;d(g)=t(g)=x\\},$

and the isotropy bundle is

$G^{\prime}:=\\{g\in G\;|\;d(g)=t(g)\\}=\bigcup_{x\in G^{(0)}}G_{x}^{x}.$

A groupoid $G$ is said to be principal if all isotropy groups are trivial, or
equivalently, $G^{\prime}=G^{(0)}$. We say that $G$ is effective if the
interior of $G^{\prime}$ equals $G^{(0)}$.

######  Definition 2.3.

Let $E$ be a finite directed graph with no sources, and let $G$ be a discrete
groupoid with unit space $G^{(0)}=E^{0}$. A self-similar action $(G,E)$ of $G$
on the path space of $E$ is given by a faithful homomorphism $G\to$
PIso$(T_{E})$ such that for every $g\in G$ and $e\in d(g)E^{1}$ there exists a
unique $h\in G$ denoted also by $g|_{e}$ and called the restriction of $g$ to
$e$ such that

$g\cdot(e\mu)=(g\cdot e)(h\cdot\mu)\;\;\text{for all}\;\;\mu\in s(e)E^{*}.$

######  Remark 2.4.

We have

$d(g|_{e})=s(e),\;t(g|_{e})=s(g\cdot e)=g|_{e}\cdot s(e),\;r(g\cdot e)=g\cdot
r(e).$

This can be visualized as

$e$$g\cdot e$$r(e)=d(g)$$g\cdot r(e)$$g$$s(e)$$s(g\cdot e)$$g|_{e}$

In particular, in general $s(g\cdot e)\neq g\cdot s(e)$, i.e. the source map
is not $G$-equivariant as it is assumed for group actions in [6]. It is shown
in Appendix A of [8] that a self-similar group action $(H,E)$ as in [6]
determines a self-similar groupoid action $(H\times E^{0},E)$ as in Definition
2.3, where $H\times E^{0}$ is the action groupoid of the group $H$. But not
any self-similar groupoid action comes from a self-similar group action.

It is possible that $g|_{e}=g$ for all $e\in d(g)E^{1}$, in which case

$g\cdot(e_{1}e_{2}\cdots e_{n})=(g\cdot e_{1})\cdots(g\cdot e_{n}).$

A self-similar groupoid action $(G,E)$ extends to an action of $G$ on the path
space $E^{*}$ and determines an action of $G$ on the graph $T_{E}$, in the
sense that $G$ acts on both the vertex space $T_{E}^{0}$ and the edge space
$T_{E}^{1}$ and intertwines the range and the source maps of $T_{E}$, see
Definition 4.1 in [4]. A self-similar groupoid action also extends to an
action of $G$ on $E^{\infty}$ such that $g\cdot\mu=\eta$ if for all $n$ we
have $g\cdot(\mu_{1}\cdots\mu_{n})=\eta_{1}\cdots\eta_{n}$.

The faithfulness condition ensures that for each $g\in G$ and $\mu\in E^{*}$
with $d(g)=r(\mu)$, there is a unique element $g|_{\mu}\in G$ satisfying

$g\cdot(\mu\nu)=(g\cdot\mu)(g|_{\mu}\cdot\nu)\;\text{ for all}\;\nu\in
s(\mu)E^{*}.$

By Proposition 3.6 of [8], self-similar groupoid actions have the following
properties: for $g,h\in G,\mu\in d(g)E^{*}$, and $\nu\in s(\mu)E^{*}$,

(1) $g|_{\mu\nu}=(g|_{\mu})|_{\nu}$;

(2) $id_{r(\mu)}|_{\mu}=id_{s(\mu)}$;

(3) if $(h,g)\in G^{(2)}$, then $(h|_{g\cdot\mu},g|_{\mu})\in G^{(2)}$ and
$(hg)|_{\mu}=(h|_{g\cdot\mu})(g|_{\mu})$;

(4) $g^{-1}|_{\mu}=(g|_{g^{-1}\cdot\mu})^{-1}$.

######  Definition 2.5.

A self-similar action $(G,E)$ is said to be level transitive if the induced
action on $E^{*}$ is transitive on each $E^{n}$. The action is level
transitive iff it is minimal on the infinite path space $E^{\infty}$.

######  Example 2.6.

Let $E$ be the graph from Example 2.1

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle
e_{2}$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{6}$$\scriptstyle e_{1}$

with $E^{0}=\\{u,v,w\\},E^{1}=\\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\\}$.
Consider the groupoid $G$ with unit space $G^{(0)}=\\{u,v,w\\}$ and generators
$a,b,c$ such that $d(a)=u,\;t(a)=v=d(b)=t(c),\;t(b)=w=d(c)$ as in the picture

$\scriptstyle u$$\scriptstyle v$$\scriptstyle w$$\scriptstyle a$$\scriptstyle
b$$\scriptstyle c$

Define the self-similar action $(G,E)$ given by

$a\cdot e_{1}=e_{2},\;\;a|_{e_{1}}=u,\;\;a\cdot e_{3}=e_{6},\;\;a|_{e_{3}}=b,$
$b\cdot e_{2}=e_{5},\;\;b|_{e_{2}}=a,\;\;b\cdot e_{6}=e_{4},\;\;b|_{e_{6}}=c,$
$c\cdot e_{4}=e_{2},\;\;c|_{e_{4}}=a^{-1},\;\;c\cdot
e_{5}=e_{6},\;\;c|_{e_{5}}=b.$

The actions of $a^{-1},b^{-1},c^{-1}$ and their restrictions are then uniquely
determined:

$a^{-1}\cdot e_{2}=e_{1},\;\;a^{-1}|_{e_{2}}=u,\;\;a^{-1}\cdot
e_{6}=e_{3},\;\;a^{-1}|_{e_{6}}=b^{-1},$ $b^{-1}\cdot
e_{5}=e_{2},\;\;b^{-1}|_{e_{5}}=a^{-1},\;\;b^{-1}\cdot
e_{4}=e_{6},\;\;b^{-1}|_{e_{4}}=c^{-1},$ $c^{-1}\cdot
e_{2}=e_{4},\;\;c^{-1}|_{e_{2}}=a,\;\;c^{-1}\cdot
e_{6}=e_{5},\;\;c^{-1}|_{e_{6}}=b^{-1}.$

The actions of $u,v,w$ and their restrictions are given by

$u\cdot e_{1}=e_{1},\;u|_{e_{1}}=u,\;u\cdot
e_{3}=e_{3},\;u|_{e_{3}}=v,\;v\cdot e_{2}=e_{2},\;v|_{e_{2}}=u,$ $v\cdot
e_{6}=e_{6},\;v|_{e_{6}}=w,\;w\cdot e_{4}=e_{4},\;w|_{e_{4}}=v,\;w\cdot
e_{5}=e_{5},\;w|_{e_{5}}=v.$

We can also characterize the self-similar action $(G,E)$ by the formulas

$a\cdot e_{1}\mu=e_{2}\mu,\;\;a\cdot
e_{3}\mu=e_{6}(b\cdot\mu),\;\;b\cdot(e_{2}\mu)=e_{5}(a\cdot\mu),$ $b\cdot
e_{6}\mu=e_{4}(c\cdot\mu),\;\;c\cdot e_{4}\mu=e_{2}(a^{-1}\cdot\mu),\;\;c\cdot
e_{5}\mu=e_{6}(b\cdot\mu)$

where $\mu\in E^{*}$, and these determine uniquely an action of $G$ on the
graph $T_{E}$.

Note that the action of $G$ is level transitive. It can be shown (see [3])
that $G$ is a transitive groupoid with isotropy $\mathbb{Z}$ and therefore
$C^{*}(G)\cong M_{3}\otimes C(\mathbb{T})$.

We observe that the graph $E$ itself does not have many symmetries. If a group
acts on $E$ by preserving the range and source maps, then all vertices must be
fixed, and the action should also fix all edges except that it could
interchange $e_{4}$ and $e_{5}$, so Aut$(E)\cong\mathbb{Z}_{2}$. The set of
self-similar groupoid actions $(G,E)$ is much richer.

######  Example 2.7.

Let $E$ be the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{3}$$\scriptstyle
e_{1}$$\scriptstyle e_{2}$$\scriptstyle e_{4}$

with $E^{0}=\\{u,v\\},E^{1}=\\{e_{1},e_{2},e_{3},e_{4}\\}$ and forest $T_{E}$

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{1}$$\scriptstyle
e_{2}$$\scriptstyle e_{1}e_{1}$$\vdots$$\scriptstyle
e_{1}e_{2}$$\vdots$$\scriptstyle e_{2}e_{2}$$\vdots$$\scriptstyle
e_{2}e_{1}$$\vdots$$\scriptstyle e_{3}$$\scriptstyle e_{4}$$\scriptstyle
e_{3}e_{1}$$\vdots$$\scriptstyle e_{3}e_{2}$$\vdots$$\scriptstyle
e_{4}e_{3}$$\vdots$$\scriptstyle e_{4}e_{4}$$\vdots$

Consider the groupoid $G=\langle a,b,c\rangle$ with
$d(a)=d(b)=d(c)=u=t(a)=t(b)$ and $t(c)=v$ as in the picture

$\scriptstyle u$$\scriptstyle v$$\scriptstyle c$$\scriptstyle a$$\scriptstyle
b$

and the self-similar action

$a\cdot e_{1}=e_{2},\;a|_{e_{1}}=b,\;a\cdot e_{2}=e_{1},\;a|_{e_{2}}=a,$
$b\cdot e_{1}=e_{1},\;b|_{e_{1}}=b,\;b\cdot e_{2}=e_{2},\;b|_{e_{2}}=a,$
$c\cdot e_{1}=e_{3},\;c|_{e_{1}}=a,\;c\cdot e_{2}=e_{4},\;c|_{e_{2}}=c.$

Then $G_{u}^{u}=\langle a,b\rangle$ is isomorphic to the lamplighter group
$L=(\mathbb{Z}_{2})^{\mathbb{Z}}\rtimes\mathbb{Z}$ and $C^{*}(G)\cong
M_{2}\otimes C^{*}(L)$.

######  Example 2.8.

Let $E$ be the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{3}$$\scriptstyle e_{1}$$\scriptstyle
e_{2}$$\scriptstyle e_{6}$$\scriptstyle e_{7}$

with $E^{0}=\\{u,v\\},E^{1}=\\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\\}$
and forest $T_{E}$

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{1}$$\scriptstyle
e_{2}$$\scriptstyle e_{3}$$\scriptstyle e_{1}e_{1}$$\vdots$$\scriptstyle
e_{1}e_{2}$$\vdots$$\vdots$$\scriptstyle e_{1}e_{3}$$\scriptstyle
e_{2}e_{1}$$\vdots$$\vdots$$\scriptstyle e_{2}e_{2}$$\scriptstyle
e_{2}e_{3}$$\vdots$$\scriptstyle e_{3}e_{4}$$\scriptstyle
e_{3}e_{5}$$\scriptstyle e_{3}e_{6}$$\scriptstyle
e_{3}e_{7}$$\vdots$$\vdots$$\vdots$$\vdots$$\scriptstyle e_{4}$$\scriptstyle
e_{5}$$\scriptstyle e_{6}$$\scriptstyle e_{7}$$\scriptstyle
e_{4}e_{1}$$\vdots$$\scriptstyle e_{4}e_{2}$$\vdots$$\scriptstyle
e_{4}e_{3}$$\vdots$$\scriptstyle e_{5}e_{1}$$\vdots$$\scriptstyle
e_{5}e_{2}$$\vdots$$\scriptstyle e_{5}e_{3}$$\vdots$$\scriptstyle
e_{6}e_{4}$$\scriptstyle e_{6}e_{5}$$\scriptstyle e_{6}e_{6}$$\scriptstyle
e_{6}e_{7}$$\vdots$$\vdots$$\vdots$$\vdots$$\scriptstyle
e_{7}e_{4}$$\scriptstyle e_{7}e_{5}$$\scriptstyle e_{7}e_{6}$$\scriptstyle
e_{7}e_{7}$$\vdots$$\vdots$$\vdots$$\vdots$

The groupoid $G$ with generators $a,b$ such that $d(a)=t(a)=u,\;d(b)=t(b)=v$
acts on $E^{*}$ by

$a\cdot e_{1}=e_{2},\;\;a|_{e_{1}}=u,\;\;a\cdot
e_{2}=e_{1},\;\;a|_{e_{2}}=a,\;\;a\cdot e_{3}=e_{3},\;\;a|_{e_{3}}=v,$ $b\cdot
e_{4}=e_{4},\;\;b|_{e_{4}}=a,\;\;b\cdot e_{5}=e_{5},\;\;b|_{e_{5}}=a,$ $b\cdot
e_{6}=e_{7},\;\;b|_{e_{6}}=v,\;\;b\cdot e_{7}=e_{6},\;\;b|_{e_{7}}=b.$

It follows that $G$ is a group bundle with $G_{u}^{u}\cong
G_{v}^{v}\cong\mathbb{Z}$, so $C^{*}(G)\cong C(\mathbb{T})\oplus
C(\mathbb{T})$. This self-similar action is related in [8] and [6] to the
Katsura algebra $\mathcal{O}_{A,B}$ with matrices

$A=\left(\begin{array}[]{cc}2&1\\\
2&2\end{array}\right),\;\;B=\left(\begin{array}[]{cc}1&0\\\
2&1\end{array}\right).$

######  Definition 2.9.

A self similar action $(G,E)$ is called contracting if there is a finite
subset $N$ of $G$ such that for every $g\in G$ there is $n\geq 0$ such that
$g|_{\mu}\in N$ for every $\mu\in d(g)E^{*}$ of length $|\mu|\geq n$. The
smallest such finite subset of $G$ is called the nucleus of the groupoid,
denoted $\mathcal{N}$.

The Moore diagram for $\mathcal{N}$ is the labelled directed graph with vertex
set $\mathcal{N}$ and, for each $g\in\mathcal{N}$ and $e\in d(g)E^{1}$, an
edge from $g\in\mathcal{N}$ to $g|_{e}\in\mathcal{N}$ labelled $(e,g\cdot e)$.
A typical edge in a Moore diagram looks like

$(e,g\cdot e)$$g$$g|_{e}$

The self-similarity relations for the set $\mathcal{N}$ can be read off the
Moore diagram: the edge above encodes the relation $g\cdot(e\mu)=(g\cdot
e)(g|_{e}\cdot\mu)$ for $\mu\in s(e)E^{*}$.

######  Example 2.10.

Consider the self-similar groupoid action $(G,E)$, where $E$ is the graph

$\scriptstyle u$$\scriptstyle v$$\scriptstyle e_{4}$$\scriptstyle
e_{1}$$\scriptstyle e_{3}$$\scriptstyle e_{2}$

and $G=\langle a,b\rangle$ where

$a\cdot e_{1}=e_{4},\;a|_{e_{1}}=u,\;a\cdot e_{2}=e_{3},\;a|_{e_{2}}=b,$
$b\cdot e_{3}=e_{1},\;b|_{e_{3}}=u,\;b\cdot e_{4}=e_{2},\;b|_{e_{4}}=a.$

Then $(G,E)$ contracts to $\mathcal{N}=\\{u,v,a,a^{-1},b,b^{-1}\\}$ and the
corresponding Moore diagram is

$\scriptstyle a$$\scriptstyle b$$\scriptstyle u$$\scriptstyle v$$\scriptstyle
a^{-1}$$\scriptstyle
b^{-1}$$\scriptstyle(e_{2},e_{3})$$\scriptstyle(e_{4},e_{2})$$\scriptstyle(e_{4},e_{4})$$\scriptstyle(e_{3},e_{3})$$\scriptstyle(e_{2},e_{2})$$\scriptstyle(e_{3},e_{1})$$\scriptstyle(e_{1},e_{4})$$\scriptstyle(e_{1},e_{3})$$\scriptstyle(e_{4},e_{1})$$\scriptstyle(e_{1},e_{1})$$\scriptstyle(e_{3},e_{2})$$\scriptstyle(e_{2},e_{4})$

see Proposition 9.2 in [8].

##  3\. Groupoids and $C^{*}$-algebras associated with self-similar actions

We first recall some constructions and results from [3].

######  Definition 3.1.

Given a self-similar groupoid action $(G,E)$, the $C^{*}$-algebra $C^{*}(G,E)$
is defined as the Cuntz-Pimsner algebra of the $C^{*}$-correspondence

$\mathcal{M}=\mathcal{M}(G,E)=\mathcal{X}(E)\otimes_{C(E^{0})}C^{*}(G)$

over $C^{*}(G)$. Here $\mathcal{X}(E)=C(E^{1})$ is the $C^{*}$-correspondence
over $C(E^{0})$ associated to the graph $E$ and $C(E^{0})=C(G^{(0)})\subseteq
C^{*}(G)$. The right action of $C^{*}(G)$ on $\mathcal{M}$ is the usual one
and the left action is determined by the representation

$W:G\to\mathcal{L}(\mathcal{M}),\;\;W_{g}(i_{e}\otimes
a)=\begin{cases}i_{g\cdot e}\otimes i_{g|_{e}}a\;\;\text{if}\;d(g)=r(e)\\\
0\;\;\text{otherwise,}\end{cases}$

where $i_{e}\in C(E^{1})$ and $i_{g}\in C_{c}(G)$ are point masses for $e\in
E^{1},g\in G$ and $a\in C^{*}(G)$. The inner product of $\mathcal{M}$ is given
by

$\langle\xi\otimes a,\eta\otimes
b\rangle=\langle\langle\eta,\xi\rangle\;a,b\rangle=a^{*}\langle\xi,\eta\rangle\;b$

for $\xi,\eta\in C(E^{1})$ and $a,b\in C^{*}(G)$.

######  Remark 3.2.

The elements $i_{e}\otimes 1$ for $e\in E^{1}$ form a Parseval frame for
$\mathcal{M}$ and every $\zeta\in\mathcal{M}$ is a finite sum

$\zeta=\sum_{e\in E^{1}}i_{e}\otimes\langle i_{e}\otimes 1,\zeta\rangle.$

In particular, if $\mathcal{X}(E)^{*}$ denotes the dual
$C^{*}$-correspondence, then

$\mathcal{L}(\mathcal{M})=\mathcal{K}(\mathcal{M})\cong\mathcal{X}(E)\otimes_{C(E^{0})}C^{*}(G)\otimes_{C(E^{0})}\mathcal{X}(E)^{*}\cong
M_{n}\otimes C^{*}(G),$

where $n=|E^{1}|$. The isomorphism is given by

$i_{e_{j}}\otimes i_{g}\otimes i_{e_{k}}^{*}\mapsto e_{jk}\otimes i_{g}$

for $E^{1}=\\{e_{1},...,e_{n}\\}$ and for matrix units $e_{jk}\in M_{n}$.
There is a unital homomorphism
$\mathcal{K}(\mathcal{X}(E))\to\mathcal{K}(\mathcal{M})$ given by

$i_{e}\otimes i_{f}^{*}\to i_{e}\otimes 1\otimes i_{f}^{*}.$

We recall the following result, see Propositions 4.4 and 4.7 in [8].

######  Theorem 3.3.

If $U_{g},P_{v}$ and $S_{e}$ are the images of $g\in G,v\in E^{0}=G^{(0)}$ and
of $e\in E^{1}$ in the Cuntz-Pimsner algebra $C^{*}(G,E)$, then

  * •

$g\mapsto U_{g}$ is a representation by partial isometries of $G$ with
$U_{v}=P_{v}$ for $v\in E^{0}$;

  * •

$S_{e}$ are partial isometries with $S_{e}^{*}S_{e}=P_{s(e)}$ and
$\displaystyle\sum_{r(e)=v}S_{e}S_{e}^{*}=P_{v}$;

  * •

$U_{g}S_{e}=\begin{cases}S_{g\cdot e}U_{g|_{e}}\;\mbox{if}\;d(g)=r(e)\\\
0,\;\mbox{otherwise;}\end{cases}$

  * •

$U_{g}P_{v}=\begin{cases}P_{g\cdot v}U_{g}\;\mbox{if}\;d(g)=v\\\
0,\;\mbox{otherwise.}\;\end{cases}$

There is a gauge action $\gamma$ of $\mathbb{T}$ on $C^{*}(G,E)$ such that
$\gamma_{z}(U_{g})=U_{g},$ and $\gamma_{z}(S_{e})=zS_{e}$ for
$z\in\mathbb{T}$.

Given $\mu=e_{1}\cdots e_{n}\in E^{*}$ with $e_{i}\in E^{1}$, we let
$S_{\mu}:=S_{e_{1}}\cdots S_{e_{n}}$. Then $C^{*}(G,E)$ is the closed linear
span of elements $S_{\mu}U_{g}S_{\nu}^{*}$, where $\mu,\nu\in E^{*}$ and $g\in
G_{s(\nu)}^{s(\mu)}$.

######  Remark 3.4.

The graph $C^{*}$-algebra $C^{*}(E)$ is embedded in $C^{*}(G,E)$ by
considering elements $S_{\mu}S_{\nu}^{*}=S_{\mu}U_{g}S_{\nu}^{*}$ where
$g=s(\nu)=s(\mu)$.

For each $k\geq 1$, consider $\mathcal{F}_{k}$ the closed linear span of
elements $S_{\mu}U_{g}S_{\nu}^{*}$ with $\mu,\nu\in E^{k}$ and $g\in
G_{s(\nu)}^{s(\mu)}$. Then the fixed point algebra
$\mathcal{F}(G,E):=C^{*}(G,E)^{\mathbb{T}}$ under the gauge action is
isomorphic to $\displaystyle\varinjlim\mathcal{F}_{k}$. We have

$\mathcal{F}_{k}\cong\mathcal{L}(\mathcal{M}^{\otimes
k})\cong\mathcal{X}(E)^{\otimes
k}\otimes_{C(E^{0})}C^{*}(G)\otimes_{C(E^{0})}\mathcal{X}(E)^{*\otimes k}$

using the map $S_{\mu}U_{g}S_{\nu}^{*}\mapsto i_{\mu}\otimes i_{g}\otimes
i_{\nu}^{*}$, where $i_{\mu}\in\mathcal{X}(E)^{\otimes k}=C(E^{k})$ are point
masses. The embeddings $\mathcal{F}_{k}\hookrightarrow\mathcal{F}_{k+1}$ are
given by

$\phi_{k}(i_{\mu}\otimes i_{g}\otimes
i_{\nu}^{*})=\begin{cases}\displaystyle\sum_{x\in d(g)E^{1}}i_{\mu y}\otimes
i_{g|_{x}}\otimes i_{\nu x}^{*},\;\text{if}\;g\in
G_{s(\nu)}^{s(\mu)}\;\text{and}\;g\cdot x=y\\\ 0,\;\text{
otherwise.}\end{cases}$

######  Definition 3.5.

A self-similar groupoid action $(G,E)$ is called pseudo free if for every
$g\in G$ and every $e\in d(g)E^{1}$, the condition $g\cdot e=e$ and
$g|_{e}=s(e)$ implies that $g=r(e)$.

For example, the self-similar action in Example 2.6 is pseudo free.

######  Remark 3.6.

If $(G,E)$ is pseudo free, then $g_{1}\cdot\alpha=g_{2}\cdot\alpha$ and
$g_{1}|_{\alpha}=g_{2}|_{\alpha}$ for some $\alpha\in E^{*}$ implies
$g_{1}=g_{2}$.

######  Proposition 3.7.

Given a self-similar groupoid action $(G,E)$, there is an inverse semigroup

$\mathcal{S}(G,E)=\\{(\alpha,g,\beta):\alpha,\beta\in E^{*},g\in
G_{s(\beta)}^{s(\alpha)}\\}\cup\\{0\\}$

with operations

$(\alpha,g,\beta)(\gamma,h,\zeta)=\begin{cases}(\alpha,g(h|_{h^{-1}\cdot\mu}),\zeta(h^{-1}\cdot\mu))&\;\text{if}\;\beta=\gamma\mu\\\
(\alpha(g\cdot\mu),g|_{\mu}h,\zeta)&\;\text{if}\;\gamma=\beta\mu\\\
0&\;\text{otherwise}\end{cases}$

and $(\alpha,g,\beta)^{*}=(\beta,g^{-1},\alpha)$.

######  Remark 3.8.

The inverse semigroup $\mathcal{S}(G,E)$ acts on the infinite path space
$E^{\infty}$ by partial homeomorphisms. The action of
$(\alpha,g,\beta)\in\mathcal{S}(G,E)$ on $\xi=\beta\mu\in\beta E^{\infty}$ is
given by

$(\alpha,g,\beta)\cdot\beta\mu=\alpha(g\cdot\mu)\in\alpha E^{\infty}.$

Note that

$r(g\cdot\mu)=g\cdot r(\mu)=g\cdot s(\beta)=s(\alpha),$

so the action is well defined.

######  Theorem 3.9.

If the self-similar action $(G,E)$ is pseudo free, then there is a locally
compact Hausdorff étale groupoid $\mathcal{G}(G,E)$ such that

$C^{*}(G,E)\cong C^{*}(\mathcal{G}(G,E)).$

If $G$ is amenable, then $C^{*}(G,E)$ is nuclear and $\mathcal{G}(G,E)$ is
also amenable.

###### Proof.

Consider the groupoid of germs associated with
$(\mathcal{S}(G,E),E^{\infty})$:

$\mathcal{G}(G,E)=\\{[\alpha,g,\beta;\xi]:\alpha,\beta\in E^{*},\;g\in
G^{s(\alpha)}_{s(\beta)},\;\xi\in\beta E^{\infty}\\}.$

Two germs
$[\alpha,g,\beta;\xi],[\alpha^{\prime},g^{\prime},\beta^{\prime};\xi^{\prime}]$
in $\mathcal{G}(G,E)$ are equal if and only if $\xi=\xi^{\prime}$ and there is
an idempotent $z\in\mathcal{S}(G,E)\setminus\\{0\\}$ such that $z\cdot\xi=\xi$
and $(\alpha,g,\beta)z=(\alpha^{\prime},g^{\prime},\beta^{\prime})z$. Using
the form of idempotents, we obtain that $\xi=\beta\gamma\eta$ for $\gamma\in
E^{*}$ and $\eta\in E^{\infty}$, with $r(\gamma)=s(\beta)$ and
$r(\eta)=s(\gamma)$. Moreover,

$\alpha^{\prime}=\alpha(g\cdot\gamma),\beta^{\prime}=\beta\gamma,\;\mbox{and}\;g^{\prime}=g|_{\gamma},$

so

$[\alpha,g,\beta;\beta\gamma\eta]=[\alpha(g\cdot\gamma),g|_{\gamma},\beta\gamma;\beta\gamma\eta].$

The unit space of $\mathcal{G}(G,E)$ is

$\mathcal{G}(G,E)^{(0)}=\\{[\alpha,s(\alpha),\alpha;\xi]:\xi\in\alpha
E^{\infty}\\},$

identified with $E^{\infty}$ by the map
$[\alpha,s(\alpha),\alpha;\xi]\mapsto\xi$.

The target and domain maps of the groupoid $\mathcal{G}(G,E)$ are given by

$t([\alpha,g,\beta;\beta\mu])=\alpha(g\cdot\mu),\;\;d([\alpha,g,\beta;\beta\mu])=\beta\mu.$

If two elements $\gamma_{1},\gamma_{2}\in\mathcal{G}(G,E)$ are composable,
then

$\gamma_{1}=[\alpha_{1},g_{1},\alpha_{2};\alpha_{2}(g_{2}\cdot\xi)],\;\;\gamma_{2}=[\alpha_{2},g_{2},\beta;\beta\xi]$

for some $\alpha_{1},\alpha_{2},\beta\in E^{*},\xi\in
E^{\infty},g_{1},g_{2}\in G$ and in this case

$\gamma_{1}\gamma_{2}=[\alpha_{1},g_{1}g_{2},\beta;\beta\xi].$

In particular,
$[\alpha,g,\beta;\beta\mu]^{-1}=[\beta,g^{-1},\alpha;\alpha(g\cdot\mu)]$.

The topology on $\mathcal{G}(G,E)$ is generated by the compact open bisections
of the form

$Z(\alpha,g,\beta;U)=\\{[\alpha,g,\beta;\xi]\in\mathcal{G}(G,E):\xi\in U\\},$

where $U$ is an open compact subset of $Z(\beta)=\beta E^{\infty}$. Since any
open set $U\subseteq Z(\beta)$ is a disjoint union $\displaystyle
U=\bigsqcup_{i}Z(\beta\gamma_{i})$ where $\gamma_{i}\in E^{*}$ with
$r(\gamma_{i})=s(\beta)$, we get

$Z(\alpha,g,\beta;U)=\bigsqcup_{i}Z(\alpha,g,\beta;Z(\beta\gamma_{i}))=\bigsqcup_{i}Z(\alpha(g\cdot\gamma_{i}),g|_{\gamma_{i}},\beta\gamma_{i};Z(\beta\gamma_{i})).$

It follows that the sets $Z(\alpha,g,\beta)=Z(\alpha,g,\beta;Z(\beta))$ form a
basis for the topology of $\mathcal{G}(G,E)$. In particular,

$d(Z(\alpha,g,\beta))=Z(\beta)\;\;\text{and}\;\;t(Z(\alpha,g,\beta))=Z(\alpha).$

Since $(G,E)$ is pseudo free, it follows that
$[\alpha,g,\beta;\xi]=[\alpha,g^{\prime},\beta;\xi]$ if and only if
$g=g^{\prime}$. Moreover, the groupoid $\mathcal{G}(G,E)$ is Hausdorff, see
Proposition 12.1 in [6]. Using the properties given in Theorem 3.3 and the
groupoid multiplication, the isomorphism $\phi:C^{*}(G,E)\to
C^{*}(\mathcal{G}(G,E))$ is given by

$\phi(P_{v})=\chi_{Z(v,v,v)},$ $\phi(T_{e})=\chi_{Z(e,s(e),s(e))},$
$\phi(U_{g})=\chi_{Z(t(g),g,d(g))}$

for $v\in E^{0},e\in E^{1}$ and $g\in G$. Here $\chi_{Z}$ is the indicator
function of $Z$.

Since $G$ is amenable, it follows that $C^{*}(G)$ is nuclear and hence the
Cuntz-Pimsner algebra $C^{*}(G,E)$ is nuclear. The same kind of argument as in
[6] shows that $\mathcal{G}(G,E)$ is also amenable. ∎

######  Remark 3.10.

Note that the graph groupoid $\mathcal{G}_{E}$ is isomorphic to an open
subgroupoid of $\mathcal{G}(G,E)$ with the same unit space $E^{\infty}$. It is
obtained as

$\mathcal{G}_{E}=\\{(\alpha\xi,|\alpha|-|\beta|,\beta\xi)\in
E^{\infty}\times\mathbb{Z}\times E^{\infty}:\alpha,\beta\in
E^{*},\;s(\alpha)=s(\beta)\\}\cong$
$\cong\\{[\alpha,s(\alpha),\beta;\beta\xi]\in\mathcal{G}(G,E):\alpha,\beta\in
E^{*},\;s(\alpha)=s(\beta),\;\xi\in E^{\infty}\\}.$

######  Definition 3.11.

Given a self-similar action $(G,E)$, we say that $E$ is $G$-transitive if
given two vertices $u,v\in E^{0}$, there are vertices
$u_{0}=u,u_{1},...,u_{2n}=v$ such that we can connect each $u_{2k-2}$ to
$u_{2k-1}$ by a path in $E$ and there are $g_{k}\in G$ with
$d(g_{k})=u_{2k-1}$ and $t(g_{k})=u_{2k}$ for all $k=1,...,n$.

Using the same method as in Theorem 13.6 and Corollary 13.7 in [6], it follows
that

######  Theorem 3.12.

Given a self-similar action $(G,E)$, the groupoid $\mathcal{G}(G,E)$ is
minimal iff $E$ is $G$-transitive. In particular, $\mathcal{G}(G,E)$ is
minimal if $G$ is transitive or if the adjacency matrix $A_{E}$ of $E$ is
irreducible. The groupoid $\mathcal{G}(G,E)$ is also minimal if the action of
$G$ is level transitive, see Definition 2.5.

Recall that a circuit (also called a cycle or a loop) in $E$ is a path
$\alpha\in E^{*}$ with $|\alpha|\geq 1$ such that $s(\alpha)=r(\alpha)$. A
$G$-circuit in $E$ is a pair $(g,\alpha)$ with $g\in G$ and $\alpha\in
d(g)E^{*}$ with $|\alpha|\geq 1$ such that $s(\alpha)=g\cdot r(\alpha)$.

Note that a circuit $\alpha$ in $E$ may be concatenated to produce an infinite
path $\xi=\alpha\alpha\alpha\cdots$ in $E^{\infty}$. If
$z=(\alpha,s(\alpha),s(\alpha))\in\mathcal{S}(G,E)$, then $z\cdot\xi=\xi$
since $r(\alpha)=s(\alpha)$.

To create fixed points from $G$-circuits $(g,\alpha)$, define a sequence
$\\{\alpha^{n}\\}_{n\geq 1}$ of finite paths and a sequence
$\\{g_{n}\\}_{n\geq 1}$ of groupoid elements recursively by
$\alpha^{1}=\alpha,g_{1}=g$ and $\alpha^{n+1}=g_{n}\cdot\alpha^{n}$ where
$g_{n+1}=g_{n}|_{\alpha^{n}}$ for $n\geq 1$. Then

$r(\alpha^{2})=r(g\cdot\alpha)=g\cdot r(\alpha)=s(\alpha)$

and assuming $r(\alpha^{k})=s(\alpha^{k-1})$ for some $k\geq 2$, we get

$r(\alpha^{k+1})=r(g_{k}\cdot\alpha^{k})=g_{k}\cdot r(\alpha^{k})=g_{k}\cdot
s(\alpha^{k-1})=$ $=g_{k-1}|_{\alpha^{k-1}}\cdot
s(\alpha^{k-1})=s(g_{k-1}\cdot\alpha^{k-1})=s(\alpha^{k}).$

The concatenation $\xi=\alpha^{1}\alpha^{2}\alpha^{3}\cdots$ is an infinite
path such that $g\cdot\xi=\alpha^{2}\alpha^{3}\alpha^{4}\cdots$. For any path
$\beta\in E^{*}$ with $s(\beta)=r(\alpha)$, the infinite path $\beta\xi$ is
fixed by $(\beta\alpha,g,\beta)\in\mathcal{S}(G,E)$ because

$(\beta\alpha,g,\beta)\cdot\beta\xi=\beta\alpha(g\cdot\xi)=\beta\xi.$

######  Remark 3.13.

Given $z=(\alpha,g,\beta)\in\mathcal{S}(G,E)$ with $|\alpha|>|\beta|$, then
$z$ admits at most one fixed point in $E^{\infty}$ and assuming
$z\cdot\xi=\xi$, then there is a $G$-circuit $(g,\gamma)$ such that
$\alpha=\beta\gamma$ and $\xi=\beta\zeta$ with $\zeta$ constructed from
$(g,\gamma)$.

Indeed, if $z\cdot\xi=\xi$, then $\xi\in Z(\beta)$ so $\xi=\beta\zeta$. Then

$\beta\zeta=\xi=(\alpha,g,\beta)(\beta\zeta)=\alpha(g\cdot\zeta)$

and since $|\alpha|>|\beta|$, we must have $\alpha=\beta\gamma$ such that
$g\cdot r(\gamma)=g\cdot s(\beta)=s(\alpha)=s(\gamma)$, i.e. $(g,\gamma)$ is a
G-circuit. Moreover, $\beta\zeta=\beta\gamma(g\cdot\zeta)$, so
$\zeta=\gamma(g\cdot\zeta)$. If we write
$\zeta=\gamma^{1}\gamma^{2}\gamma^{3}\cdots$ with $|\gamma^{i}|=|\gamma|$ for
$i\geq 1$, then

$\gamma^{1}\gamma^{2}\gamma^{3}\cdots=\gamma(g\cdot\zeta)=\gamma(g\cdot\gamma^{1})(g|_{\gamma^{1}}\cdot\gamma^{2})(g|_{\gamma^{1}}|_{\gamma^{2}}\cdot\gamma^{3})\cdots.$

It follows that $\gamma^{1}=\gamma$ and $\gamma^{n+1}=g_{n}\cdot\gamma^{n}$
where $g_{1}=g$ and $g_{n+1}=g_{n}|_{\gamma^{n}}$ for $n\geq 1$.

Using the methods of Theorem 14.10 and Corollary 14.13 in [6] to prove that
the interior of the isotropy of $\mathcal{G}(G,E)$ is $E^{\infty}$, we obtain

######  Theorem 3.14.

Suppose that we have a pseudo free self-similar action $(G,E)$. Then the
groupoid $\mathcal{G}(G,E)$ is effective if and only if

(a) every $G$-circuit has an entry (which is the same as saying that every
circuit has an entry, since $E$ is finite);

(b) for every $g\in G\setminus G^{(0)}$ there is $\zeta\in Z(d(g))$ such that
$g\cdot\zeta\neq\zeta$.

######  Corollary 3.15.

If $\mathcal{G}(G,E)$ is Hausdorff, effective and minimal, then
$\mathcal{G}(G,E)$ is purely infinite.

###### Proof.

This follows since the open ample subgroupoid $\mathcal{G}_{E}$ of
$\mathcal{G}(G,E)$ is purely infinite. ∎

For example, the groupoid $\mathcal{G}(G,E)$ obtained from Example 2.6 is
minimal and effective. In particular, its $C^{*}$-algebra is simple and purely
infinite.

##  4\. Tables and the Higman-Thompson groups

Consider a finite directed graph $E$ with no sources such that $E^{\infty}$ is
a Cantor set. We recall first some facts about the Higman-Thompson group of
$E$.

######  Definition 4.1.

A table over the graph $E$ is a matrix of the form

$\tau=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right),$

where $\alpha_{i},\beta_{i}\in E^{*}$ are such that
$s(\alpha_{i})=s(\beta_{i})$ and $E^{\infty}$ is decomposed into disjoint
unions

$E^{\infty}=\bigsqcup_{i=1}^{m}Z(\alpha_{i})=\bigsqcup_{j=1}^{m}Z(\beta_{j}),$

i.e. for every infinite path $\mu\in E^{\infty}$ exactly one $\alpha_{i}$ and
exactly one $\beta_{j}$ is a prefix of $\mu$.

It follows that the finite paths in each row of a table are incomparable (none
of them is a begining of the other) and the corresponding cylinder sets
$Z(\alpha_{i})$ and $Z(\beta_{j})$ are indeed disjoint. Every table $\tau$
defines a homeomorphism $\bar{\tau}$ of $E^{\infty}$ by the rule

$\bar{\tau}(\beta_{i}\mu)=\alpha_{i}\mu,\;\text{ for }\;i=1,...,m.$

This homeomorphism does not change if we permute the columns of $\tau$. Its
inverse is given by

$\tau^{-1}=\left(\begin{array}[]{cccc}\beta_{1}&\beta_{2}&\cdots&\beta_{m}\\\
\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\end{array}\right).$

A split of a table is a matrix obtained from the original one by several
replacements of a column $\left(\begin{array}[]{c}\alpha_{i}\\\
\beta_{i}\end{array}\right)$ by a matrix

$\left(\begin{array}[]{cccc}\alpha_{i}e_{1}&\alpha_{i}e_{2}&\cdots&\alpha_{i}e_{d}\\\
\beta_{i}e_{1}&\beta_{i}e_{2}&\cdots&\beta_{i}e_{d}\end{array}\right),$

where $s(\alpha_{i})E^{1}=\\{e_{1},e_{2},...,e_{d}\\}=s(\beta_{i})E^{1}$. It
is easy to check that a split of a table is also a table, and two tables
$\tau_{1},\tau_{2}$ define the same homeomorphisms of $E^{\infty}$ if they
have splits which agree up to a permutation of the columns.

The set of all homeomorphisms of $E^{\infty}$ defined by such tables is a
subgroup of Homeo$(E^{\infty})$, called the Higman-Thompson group of $E$ and
denoted by $V_{E}$. In [9], the authors define tables associated to a one-
sided topological Markov shift $(X_{A},\sigma_{A})$ for an irreducible
$N\times N$ matrix $A$ with entries in $\\{0,1\\}$. They studied the
corresponding Higman-Thompson group and proved that it is isomorphic to the
topological full group of the groupoid associated to $(X_{A},\sigma_{A})$.
Their results were generalized for graph groupoids in [18].

######  Remark 4.2.

The correspondence

$\tau=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right)\mapsto
T=S_{\alpha_{1}}S_{\beta_{1}}^{*}+S_{\alpha_{2}}S_{\beta_{2}}^{*}+\cdots+S_{\alpha_{m}}S_{\beta_{m}}^{*}$

defines a faithful unitary representation of the group $V_{E}$ in the
$C^{*}$-algebra of the graph $E$. Moreover,

$T\in N(C(E^{\infty}),C^{*}(E))=\\{u\in
U(C^{*}(E)):uC(E^{\infty})u^{*}=C(E^{\infty})\\}$

and $f\circ\bar{\tau}^{-1}=TfT^{*}$ for $f\in C(E^{\infty})$.

###### Proof.

Indeed, as in [9],
$T=S_{\alpha_{1}}S_{\beta_{1}}^{*}+S_{\alpha_{2}}S_{\beta_{2}}^{*}+\cdots+S_{\alpha_{m}}S_{\beta_{m}}^{*}$
is a unitary in $C^{*}(E)$ since $\displaystyle
E^{\infty}=\bigsqcup_{i=1}^{m}Z(\alpha_{i})=\bigsqcup_{i=1}^{m}Z(\beta_{i})$
and therefore

$T^{*}T=(\sum_{i=1}^{m}S_{\beta_{i}}S_{\alpha_{i}}^{*})(\sum_{j=1}^{m}S_{\alpha_{j}}S_{\beta_{j}}^{*})=\sum_{i=1}^{m}S_{\beta_{i}}S_{\beta_{i}}^{*}=\sum_{v\in
E^{0}}P_{v}=I,$
$TT^{*}=(\sum_{i=1}^{m}S_{\alpha_{i}}S_{\beta_{i}}^{*})(\sum_{j=1}^{m}S_{\beta_{j}}S_{\alpha_{j}}^{*})=\sum_{i=1}^{m}S_{\alpha_{i}}S_{\alpha_{i}}^{*}=\sum_{v\in
E^{0}}P_{v}=I.$

For $f=\chi_{Z(\eta)}=S_{\eta}S_{\eta}^{*}\in C(E^{\infty})$ a computation
shows that $T\chi_{Z(\eta)}T^{*}=\chi_{Z(\eta)}\circ\bar{\tau}^{-1}$. ∎

######  Example 4.3.

For the graph $E$ in Example 2.1, a table is

$\tau=\left(\begin{array}[]{cccc}u&e_{4}&e_{5}&v\\\
u&e_{5}&e_{4}&v\end{array}\right),$

with split

$\tau^{\prime}=\left(\begin{array}[]{ccccc}u&e_{4}e_{2}&e_{4}e_{6}&e_{5}&v\\\
u&e_{5}e_{2}&e_{5}e_{6}&e_{4}&v\end{array}\right).$

Consider now a self-similar groupoid action $(G,E)$. Assume that $|uE^{1}|=d$
is constant for all $u\in E^{0}$.

######  Definition 4.4.

A $G$-table is a matrix of the form

$\tau=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
g_{1}&g_{2}&\cdots&g_{m}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right),$

where $g_{i}\in G_{s(\beta_{i})}^{s(\alpha_{i})}$ and
$E^{\infty}=\bigsqcup_{i=1}^{m}Z(\alpha_{i})=\bigsqcup_{i=1}^{m}Z(\beta_{i})$.

A $G$-table $\tau$ determines a homeomorphism $\bar{\tau}$ of $E^{\infty}$
taking $\beta_{i}\mu$ into $\alpha_{i}(g_{i}\cdot\mu)$. This homeomorphism
does not change if we permute the columns of $\tau$. The set of all
homeomorphisms defined by such tables is a subgroup of Homeo$(E^{\infty})$,
called the Higman-Thompson group of $(G,E)$ and denoted by $V_{E}(G)$.

The inverse of $\bar{\tau}$ corresponds to the table

$\tau^{-1}=\left(\begin{array}[]{cccc}\beta_{1}&\beta_{2}&\cdots&\beta_{m}\\\
g_{1}^{-1}&g_{2}^{-1}&\cdots&g_{m}^{-1}\\\
\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\end{array}\right).$

The splitting rule for $G$-tables will replace a column
$\displaystyle\left(\begin{array}[]{c}\alpha\\\ g\\\ \beta\end{array}\right)$
by the matrix

$\left(\begin{array}[]{cccc}\alpha f_{1}&\alpha f_{2}&\cdots&\alpha f_{d}\\\
h_{1}&h_{2}&\cdots&h_{d}\\\ \beta e_{1}&\beta e_{2}&\cdots&\beta
e_{d}\end{array}\right),$

where
$s(\alpha)E^{1}=\\{f_{1},f_{2},...,f_{d}\\},s(\beta)E^{1}=\\{e_{1},e_{2},...,e_{d}\\}$,
$g\cdot e_{j}=f_{j}$ and $h_{j}=g|_{e_{j}}$ for all $j=1,...,d$. It is easy to
check that a split of a table is also a table, and two tables
$\tau_{1},\tau_{2}$ define the same homeomorphisms of $E^{\infty}$ if they
have splits which agree up to a permutation of the columns.

######  Lemma 4.5.

Given a $G$-table
$\displaystyle\tau=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
g_{1}&g_{2}&\cdots&g_{m}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right)$, the correspondence

$\tau\mapsto
T=S_{\alpha_{1}}U_{g_{1}}S_{\beta_{1}}^{*}+S_{\alpha_{2}}U_{g_{2}}S_{\beta_{2}}^{*}+\cdots+S_{\alpha_{m}}U_{g_{m}}S_{\beta_{m}}^{*}$

defines a faithful unitary representation of the group $V_{E}(G)$ in the
$C^{*}$-algebra $C^{*}(G,E)$.

###### Proof.

This follows from the computations

$(\sum_{i=1}^{m}S_{\alpha_{i}}U_{g_{i}}S_{\beta_{i}}^{*})(\sum_{j=1}^{m}S_{\beta_{j}}U^{-1}_{g_{j}}S_{\alpha_{i}}^{*})=\sum_{i=1}^{m}S_{\alpha_{i}}S_{\alpha_{i}}^{*}=\sum_{v\in
E^{0}}P_{v}=I,$
$(\sum_{i=1}^{m}S_{\beta_{i}}U_{g_{i}}^{-1}S_{\alpha_{i}}^{*})(\sum_{j=1}^{m}S_{\alpha_{j}}U_{g_{j}}S_{\beta_{j}}^{*})=\sum_{i=1}^{m}S_{\beta_{i}}S_{\beta_{i}}^{*}=\sum_{v\in
E^{0}}P_{v}=I.$

∎

######  Remark 4.6.

The group $V_{E}$ can be identified with a subgroup of $V_{E}(G)$ such that

$\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right)\mapsto\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
s(\alpha_{1})&s(\alpha_{2})&\cdots&s(\alpha_{m})\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right).$

######  Example 4.7.

For the groupoid action in Example 2.6, a $G$-table is

$\tau=\left(\begin{array}[]{cccc}u&v&e_{4}&e_{5}\\\ a^{-1}&cb&cb&a\\\
e_{4}&e_{5}&v&u\end{array}\right)$

with split

$\tau^{\prime}=\left(\begin{array}[]{ccccc}e_{1}&e_{3}&v&e_{4}&e_{5}\\\
u&b^{-1}&cb&cb&a\\\ e_{4}e_{2}&e_{4}e_{6}&e_{5}&v&u\end{array}\right).$

##  5\. Topological full groups and the AH conjecture

The study of full groups in the setting of topological dynamics was initiated
by T. Giordano, I. F. Putnam and C. F. Skau in [7]. For a minimal action
$\varphi:\mathbb{Z}\curvearrowright X$ on a Cantor set, they defined several
types of full groups and showed that these groups completely determine the
orbit equivalence class, the strong orbit equivalence class and the flip
conjugacy class of $\varphi$, respectively. Topological full groups associated
to dynamical systems and to étale groupoids are complete invariants for
continuous orbit equivalence and groupoid isomorphism. They also provide means
of constructing new groups with interesting properties, most notably by
providing the first examples of finitely generated infinite simple groups that
are amenable.

######  Example 5.1.

For the $AF$-equivalence relation associated to a Bratteli diagram $(V,E)$
with levels $(V_{m},E_{m})$, the topological full group is an increasing union
of groups of the form $\displaystyle\bigoplus_{v\in V_{m}}S_{n(v)}$, where
$n(v)$ denotes the number of paths from the top vertex $v_{0}$ to $v\in V_{m}$
and $S_{n}$ is the symmetric group, see Proposition 3.3 in [10].

######  Definition 5.2.

A second countable groupoid $G$ is ample if it is étale and $G^{(0)}$ is zero-
dimensional; equivalently, $G$ is ample if it has a basis of compact open
bisections. An ample groupoid $G$ is elementary if it is compact and
principal. $G$ is an $AF$ groupoid if there exists an ascending chain of open
elementary subgroupoids $G_{1}\subseteq G_{2}\subseteq...\subseteq G$ such
that $G=\bigcup_{i=1}^{\infty}G_{i}$.

We recall now the definion of homology of étale groupoids which was introduced
by Crainic and Moerdijk in [2]. Let $A$ be an abelian group and let $\pi:X\to
Y$ be a local homeomorphism between two locally compact Hausdorff spaces.
Given any $f\in C_{c}(X,A)$ we define

$\pi_{*}(f)(y):=\sum_{\pi(x)=y}f(x).$

It follows that $\pi_{*}(f)\in C_{c}(Y,A)$. Given an étale groupoid $G$, let
$G^{(1)}=G$ and for $n\geq 2$ let $G^{(n)}$ be the space of composable strings
of $n$ elements in $G$ with the product topology. For $n\geq 2$ and
$i=0,...,n$, we let $\partial_{i}:G^{(n)}\to G^{(n-1)}$ be the face maps
defined by

$\partial_{i}(g_{1},g_{2},...,g_{n})=\begin{cases}(g_{2},g_{3},...,g_{n})&\;\text{if}\;i=0,\\\
(g_{1},...,g_{i}g_{i+1},...,g_{n})&\;\text{if}\;1\leq i\leq n-1,\\\
(g_{1},g_{2},...,g_{n-1})&\;\text{if}\;i=n.\end{cases},$

which are local homeomorphisms. We define the homomorphisms
$\delta_{n}:C_{c}(G^{(n)},A)\to C_{c}(G^{(n-1)},A)$ given by

$\delta_{1}=d_{*}-t_{*},\;\;\delta_{n}=\sum_{i=0}^{n}(-1)^{i}\partial_{i*}\;\text{for}\;n\geq
2.$

Recall that $d,t:G\to G^{(0)}$ are the domain and target maps. It can be
verified that $\delta_{n}\circ\delta_{n+1}=0$ for all $n\geq 1$.

The homology groups $H_{n}(G,A)$ are by definition the homology groups of the
chain complex $C_{c}(G^{(*)},A)$ given by

$0\stackrel{{\scriptstyle\delta_{0}}}{{\longleftarrow}}C_{c}(G^{(0)},A)\stackrel{{\scriptstyle\delta_{1}}}{{\longleftarrow}}C_{c}(G^{(1)},A)\stackrel{{\scriptstyle\delta_{2}}}{{\longleftarrow}}C_{c}(G^{(2)},A)\longleftarrow\cdots,$

i.e. $H_{n}(G,A)=\ker\delta_{n}/\text{im }\delta_{n+1}$, where $\delta_{0}=0$.
If $A=\mathbb{Z}$, we write $H_{n}(G)$ for $H_{n}(G,\mathbb{Z})$.

######  Example 5.3.

For the transformation groupoid $\Gamma\ltimes X$ associated to a countable
discrete group action $\Gamma\curvearrowright X$ on a Cantor set, it follows
that

$H_{n}(\Gamma\ltimes X)\cong H_{n}(\Gamma,C(X,\mathbb{Z})),$

the group homology of $\Gamma$ with coefficients in $C(X,\mathbb{Z})$.

If $U\subseteq G$ is a bisection of an étale groupoid $G$, then $t|_{U}:U\to
t(U)$ is a homeomorphism, and similarly for $d$. Thus we get a homeomorphism
$\pi_{U}:=t|_{U}\circ(d|_{U})^{-1}$ from $d(U)$ to $t(U)$ which maps $d(g)$ to
$t(g)$ for each $g\in U$. We say that the bisection $U$ is full if
$t(U)=d(U)=G^{(0)}$, and in this case $\pi_{U}$ is a homeomorphism of
$G^{(0)}$. For a homeomorphism $\alpha:X\to X$ of a topological space $X$ we
define the support of $\alpha$ to be the set

$\text{supp}(\alpha):=\overline{\\{x\in X\mid\alpha(x)\neq x\\}}.$

######  Definition 5.4.

The topological full group of an effective étale groupoid $G$ is

$\ldbrack G\rdbrack:=\\{\pi_{U}\mid U\subseteq G\;\text{full bisection}\\},$

which is a subgroup of the homeomorphism group of $G^{(0)}$ such that
$\pi_{U}\circ\pi_{V}=\pi_{UV}$. The commutator subgroup of $\ldbrack
G\rdbrack$ is denoted by $D(\ldbrack G\rdbrack)$ and the abelianization of
$\ldbrack G\rdbrack$ is $\ldbrack G\rdbrack_{ab}=\ldbrack G\rdbrack/D(\ldbrack
G\rdbrack)$.

We observe that when $G$ is effective and Hausdorff, then
$\text{supp}(\pi_{U})$ is also open for any full bisection $U$.

######  Remark 5.5.

For a second countable effective ample groupoid $G$, the group $\ldbrack
G\rdbrack$ is at most countable and there is a natural short exact sequence

$1\to U(C(G^{(0)}))\to N(C(G^{(0)}),C^{*}_{r}(G))\to\ldbrack G\rdbrack\to 1,$

where $N(C(G^{(0)}),C^{*}_{r}(G))$ denotes the group of unitary normalizers of
$C(G^{(0)})$ in $C^{*}_{r}(G)$ (Proposition 5.6 in [11]).

Given a bisection $U\subseteq G$ with $t(U)\cap d(U)=\emptyset$, construct

$\hat{U}:=U\sqcup U^{-1}\sqcup(G^{(0)}\setminus(t(U)\cup d(U))),$

which is a full bisection. The associated homeomorphism
$\pi_{\hat{U}}\in\ldbrack G\rdbrack$ satisfies

$\pi_{\hat{U}}(d(U))=t(U),\;\pi_{\hat{U}}(t(U))=d(U),$
$\text{supp}(\pi_{\hat{U}})=t(U)\cup
d(U),\;\pi_{\hat{U}}\circ\pi_{\hat{U}}=id_{G^{(0)}}.$

An element $\pi\in\ldbrack G\rdbrack$ that satisfies $\pi^{2}=id_{G^{(0)}}$ is
called a transposition.

For an effective minimal second countable Hausdorff ample groupoid $G$ with
$G^{(0)}$ a Cantor set, the AH-conjecture of Matui claims that the following
sequence is exact

$H_{0}(G)\otimes\mathbb{Z}_{2}\stackrel{{\scriptstyle
j}}{{\longrightarrow}}\ldbrack G\rdbrack_{ab}\stackrel{{\scriptstyle
I_{ab}}}{{\longrightarrow}}H_{1}(G)\to 0.$

In particular, if the group $H_{0}(G)$ is $2$-divisible, then this implies
that $\ldbrack G\rdbrack_{ab}\cong H_{1}(G)$. The map $I_{ab}$ is induced by
the index map $I:\ldbrack G\rdbrack\to H_{1}(G)$ given by
$\pi_{U}\mapsto[\chi_{U}]$, where $U$ is a full bisection of $G$ and
$\chi_{U}$ is the indicator function of $U$, which belongs to
$\ker\delta_{1}=\ker(d_{*}-t_{*})$. The map $j$ takes
$[\chi_{d(U)}]\otimes\bar{1}$ into $[\pi_{\hat{U}}]$, where $U\subseteq G$ is
a bisection with $d(U)\cap t(U)=\emptyset$ and $\pi_{\hat{U}}\in\ldbrack
G\rdbrack$ is the transposition defined above.

######  Remark 5.6.

The AH-conjecture has been verified for several cases of groupoids, but so far
no counter-example has been found. It is shown by H. Matui in [12] that if the
groupoid $G$ is purely infinite, then the index map $I:\ldbrack G\rdbrack\to
H_{1}(G)$ is surjective. Also, in this case, $\ldbrack G\rdbrack$ contains a
copy of $\mathbb{Z}_{2}\ast\mathbb{Z}_{3}$, so $\ldbrack G\rdbrack$ is not
amenable.

Denote by $\mathcal{T}(G)$ the subgroup of $\ldbrack G\rdbrack$ generated by
all transpositions. The subgroup $\mathcal{T}(G)$ is the analog of the
symmetric group. One has $\mathcal{T}(G)\subseteq\ker(I)$. Indeed, using Lemma
7.3 in [11] we have

$I(\pi_{\hat{U}})=[\chi_{\hat{U}}]=[\chi_{U\sqcup
U^{-1}\sqcup(G^{(0)}\setminus\text{supp}(\pi_{\hat{U}})}]=[\chi_{U}+\chi_{U^{-1}}]=0\in
H_{1}(G).$

We say that $G$ has property TR if $\mathcal{T}(G)=\ker(I)$. In order to
verify the AH conjecture for $G$, it suffices to establish property TR. In
general, property TR implies the inclusion
$\ker(I_{ab})\subseteq\text{im}(j)$. The converse holds if the commutator
subgroup $D(\ldbrack G\rdbrack)$ is simple, since in this case $D(\ldbrack
G\rdbrack)=\mathcal{A}(G)$, the analog of the alternating subgroup of
$\mathcal{T}(G)$ defined in [17]. The group $D(\ldbrack G\rdbrack)$ is known
to be simple for minimal groupoids which are either almost finite or purely
infinite, see [12].

We now specialize to the groupoid $\mathcal{G}(G,E)$ associated to a self-
similar action $(G,E)$ such that $E^{\infty}$ is a Cantor set. Recall that $E$
is a finite graph with no sources and $G^{(0)}=E^{0}$.

######  Theorem 5.7.

For a self-similar action $(G,E)$ such that $\mathcal{G}(G,E)$ is effective,
we have $V_{E}(G)\cong\ldbrack\mathcal{G}(G,E)\rdbrack$. In particular,
$\ldbrack\mathcal{G}_{E}\rdbrack\subseteq\ldbrack\mathcal{G}(G,E)\rdbrack$.

###### Proof.

Indeed, given a unitary element
$\displaystyle\sum_{i=1}^{m}S_{\alpha_{i}}U_{g_{i}}S_{\beta_{i}}^{*}$ of
$V_{E}(G)$ corresponding to the $G$-table
$\displaystyle\tau=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{m}\\\
g_{1}&g_{2}&\cdots&g_{m}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{m}\end{array}\right)$, this determines a
full bisection $U=\bigsqcup_{i=1}^{m}Z(\alpha_{i},g_{i},\beta_{i})$ of
$\mathcal{G}(G,E)$ and an element $\pi_{U}=\bar{\tau}$ of
$\ldbrack\mathcal{G}(G,E)\rdbrack$, so we get a map
$V_{E}(G)\to\ldbrack\mathcal{G}(G,E)\rdbrack$. It is easy to see that this map
is a group homomorphism. It is injective since if
$\displaystyle\sum_{i=1}^{m}S_{\alpha_{i}}U_{g_{i}}S_{\beta_{i}}^{*}$
determines the identity of $\mathcal{G}(G,E)^{(0)}=E^{\infty}$, then
$\alpha_{i}=\beta_{i}$ and $g_{i}=s(\alpha_{i})$ for $1\leq i\leq m$, so

$\sum_{i=1}^{m}S_{\alpha_{i}}U_{g_{i}}S_{\beta_{i}}^{*}=\sum_{v\in
E^{0}}P_{v}=I.$

To show surjectivity, given $\pi\in\ldbrack\mathcal{G}(G,E)\rdbrack$, since
$\mathcal{G}(G,E)$ is effective, there is a unique full bisection
$U=\bigsqcup_{i=1}^{m}Z(\alpha_{i},g_{i},\beta_{i})$ with $\pi=\pi_{U}$ which
determines an element
$\displaystyle\sum_{i=1}^{m}S_{\alpha_{i}}U_{g_{i}}S_{\beta_{i}}^{*}$ of
$V_{E}(G)$. ∎

######  Corollary 5.8.

Assuming that $\mathcal{G}(G,E)$ is amenable, we have an exact sequence

$1\to U(C(E^{\infty}))\to
N(C(E^{\infty}),C^{*}(G,E))\to\ldbrack\mathcal{G}(G,E)\rdbrack\to 1$

which splits, where

$N(C(E^{\infty}),C^{*}(G,E))=\\{u\in
U(C^{*}(G,E))\;\mid\;uC(E^{\infty})u^{*}=C(E^{\infty})\\}.$

The splitting is given by $U\mapsto\chi_{U}$ where
$U=\bigsqcup_{i=1}^{m}Z(\alpha_{i},g_{i},\beta_{i})$.

Questions. When is $\ldbrack\mathcal{G}(G,E)\rdbrack$ finitely generated? When
is of type $F_{\infty}$? When does it have the Haagerup property?

######  Remark 5.9.

For a pseudo free self-similar action $(G,E)$ with Exel-Pardo groupoid
$\mathcal{G}(G,E)$, there is a cocycle $\rho:\mathcal{G}(G,E)\to\mathbb{Z}$
given by $[\alpha,g,\beta;\xi]\mapsto|\alpha|-|\beta|$ with kernel

$\mathcal{H}(G,E)=\\{[\alpha,g,\beta;\xi]\in\mathcal{G}(G,E):|\alpha|=|\beta|\\}.$

The cocyle is well-defined since it does not depend on germ representatives.
Now $\mathcal{H}(G,E)=\bigcup_{k\geq 0}\mathcal{H}_{k}(G,E)$ where

$\mathcal{H}_{k}(G,E)=\\{[\alpha,g,\beta;\xi]\in\mathcal{G}(G,E):|\alpha|=|\beta|=k\\}.$

There are groupoid homomorphisms

$\omega_{k}:\mathcal{H}_{k}(G,E)\to G,\;\omega_{k}([\alpha,g,\beta;\xi])=g$

and $\ker\omega_{k}$ is $AF$ for all $k\geq 0$.

###### Proof.

Indeed, consider $R_{k}$ the equivalence relation on $E^{k}$ such that
$(\alpha,\beta)\in R_{k}$ if there is $g\in G$ with $g\cdot
s(\beta)=s(\alpha)$. Then the map
$[\alpha,g,\beta;\xi]\mapsto((\xi,g),(\alpha,\beta))$ gives an isomorphism
between $\mathcal{H}_{k}(G,E)$ and $(E^{\infty}\rtimes G)\times R_{k}$, so
$\ker\omega_{k}$ is isomorphic to $E^{\infty}\times R_{k}$. Since the
groupoids $\mathcal{H}_{k}(G,E)\times_{\omega_{k}}G$ and $\ker\omega_{k}$ are
similar, we obtain

$H_{i}(\mathcal{H}_{k}(G,E)\times_{\omega_{k}}G)\cong H_{i}(\ker\omega_{k})=0$

for $i\geq 1$. ∎

######  Corollary 5.10.

We have

$H_{i}(\mathcal{H}(G,E))=\varinjlim H_{i}(\mathcal{H}_{k}(G,E)),$

where the inclusion maps are given by

$\chi_{Z(\alpha,g,\beta;Z(\beta\gamma))}\mapsto\chi_{Z(\alpha(g\cdot\gamma),g|_{\gamma},\beta\gamma;Z(\beta\gamma))}.$

Using the notation from a previous section, we have

$C^{*}(G,E)^{\mathbb{T}}=\mathcal{F}(G,E)\cong
C^{*}(\mathcal{H}(G,E)),\;\;\mathcal{F}_{k}\cong C^{*}(\mathcal{H}_{k}(G,E)).$

We also have

$\bigcup_{k\geq
0}\ldbrack\mathcal{H}_{k}(G,E)\rdbrack=\ldbrack\mathcal{H}(G,E)\rdbrack\subseteq\ldbrack\mathcal{G}(G,E)\rdbrack$

by considering $G$-tables where $|\alpha_{i}|=|\beta_{i}|$.

######  Remark 5.11.

Assuming that the groupoid $G$ is transitive, it follows that
$\mathcal{G}(G,E)$ is minimal. Then $G$ is Morita equivalent to the group
$G_{u}^{u}$ for some $u\in E^{0}$, and $\mathcal{H}_{k}(G,E)$ is Morita
equivalent to $E^{\infty}\rtimes G$. We deduce that

$H_{q}(\mathcal{H}_{k}(G,E))\cong H_{q}(E^{\infty}\rtimes G)\cong
H_{q}(G,C(E^{\infty},\mathbb{Z}))\cong
H_{q}(G_{u}^{u},C(uE^{\infty},\mathbb{Z}))$

and

$H_{q}(\mathcal{H}(G,E))\cong\varinjlim
H_{q}(G_{u}^{u},C(uE^{\infty},\mathbb{Z})).$

Recall that the boundary map $\delta_{1}:C_{c}(\mathcal{H}(G,E))\to
C_{c}(\mathcal{H}(G,E)^{(0)})$ is given by

$\delta_{1}(\chi_{Z(\alpha,g,\beta)})=\chi_{Z(\beta,s(\beta),\beta)}-\chi_{Z(\alpha,s(\alpha),\alpha)}$

for $\alpha,\beta\in E^{*}$ with $|\alpha|=|\beta|$ and $g\cdot
s(\beta)=s(\alpha)$. It follows that

$[\chi_{Z(\alpha,s(\alpha),\alpha)}]=[\chi_{Z(\beta,s(\beta),\beta)}]\in
H_{0}(\mathcal{H}(G,E))$

whenever $(\alpha,\beta)\in R_{k}$ for some $k$. Hence for $G$ transitive, the
group $H_{0}(\mathcal{H}(G,E))$ is generated by $[\chi_{Z(\alpha,u,\alpha)}]$
where $\alpha\in E^{*}u$.

The boundary map $\delta_{2}:C_{c}(\mathcal{H}(G,E)^{(2)})\to
C_{c}(\mathcal{H}(G,E))$ is given by

$\delta_{2}(\chi_{Z(\alpha,g,\beta)\times
Z(\beta,h,\gamma)})=\chi_{Z(\beta,h,\gamma)}-\chi_{Z(\alpha,gh,\gamma)}+\chi_{Z(\alpha,g,\beta)}$

for $\alpha,\beta,\gamma\in E^{*}$ with $|\alpha|=|\beta|=|\gamma|$ and
$g\cdot s(\beta)=s(\alpha)$, $h\cdot s(\gamma)=s(\beta)$. It follows that in
$H_{1}(\mathcal{H}(G,E))=\ker\delta_{1}/\text{im }\delta_{2}$ we have

$[\chi_{Z(\alpha,g,\beta)}]+[\chi_{Z(\beta,h,\gamma)}]=[\chi_{Z(\alpha,gh,\gamma)}].$

Taking $h=d(g)=s(\beta)$ and $\gamma=\beta$ we get

$[\chi_{Z(\alpha,g,\beta)}]=[\chi_{Z(\alpha,g,\beta)}]+[\chi_{Z(\beta,s(\beta),\beta)}],$

so $[\chi_{Z(\beta,s(\beta),\beta)}]=0$ in $H_{1}(\mathcal{H}(G,E))$. Taking
$h=g^{-1}$ and $\beta=\gamma$ we get

$[\chi_{Z(\alpha,g,\beta)}]=-[\chi_{Z(\beta,g^{-1},\alpha)}].$

If $s(\alpha)=s(\beta)=u$ and $|\alpha|=|\beta|$, we obtain for $g\in
G_{u}^{u}$

$[\chi_{Z(\alpha,g,\beta)}]=[\chi_{Z(\alpha,g,\alpha)}]+[\chi_{Z(\alpha,u,\beta)}],$
$[\chi_{Z(\beta,g,\alpha)}]=[\chi_{Z(\beta,g,\beta)}]+[\chi_{Z(\beta,u,\alpha)}],$

so

$[\chi_{Z(\alpha,g,\alpha)}]=[\chi_{Z(\alpha,g,\beta)}]-[\chi_{Z(\alpha,u,\beta)}]=-[\chi_{Z(\beta,g^{-1},\alpha)}]+[\chi_{Z(\beta,u,\alpha)}]=$
$=-[\chi_{Z(\beta,g^{-1},\beta)}]-[\chi_{Z(\beta,u,\alpha)}]+[\chi_{Z(\beta,u,\alpha)}]=[\chi_{Z(\beta,g,\beta)}].$

For $g,h\in G_{s(\alpha)}^{s(\alpha)}$ we have

$[\chi_{Z(\alpha,g,\alpha)}]+[\chi_{Z(\alpha,h,\alpha)}]=[\chi_{Z(\alpha,gh,\alpha)}].$

######  Proposition 5.12.

Let $(G,E)$ be a self-similar action with $G$ transitive such that
$\mathcal{G}(G,E)$ is effective. If the homology group
$H_{1}(\mathcal{H}(G,E))$ is spanned by $[\chi_{Z(\alpha,g,\alpha)}]$ where
$g\in G_{s(\alpha)}^{s(\alpha)}$, then the index map
$I_{\mathcal{H}}:\ldbrack\mathcal{H}(G,E)\rdbrack\to H_{1}(\mathcal{H}(G,E))$
is surjective.

###### Proof.

Let $\alpha\in E^{*}$ and consider the bisection $V=Z(\alpha,g,\alpha)$ where
$g\in G_{s(\alpha)}^{s(\alpha)}$. Since $d(V)=t(V)=Z(\alpha)$, we can define a
full bisection $U=V\sqcup(E^{\infty}\setminus
Z(\alpha))\subseteq\mathcal{H}(G,E)$ and we have
$I_{\mathcal{H}}(\pi_{U})=[\chi_{V}]$ by using Lemma 7.3 in [11]. The result
follows since the elements $[\chi_{V}]$ span $H_{1}(\mathcal{H}(G,E))$. ∎

The AH conjecture was verified for the Katsura-Exel-Pardo groupoids by P.
Nyland and E. Ortega in [19]. The groupoid $\mathcal{G}(G,E)$ obtained from
Example 2.8 is one the these. One of the main ingredients of the proof is to
show the property TR by using a decomposition

$\ldbrack\mathcal{G}(G,E)\rdbrack=\ldbrack\mathcal{H}(G,E)\rdbrack\ldbrack\mathcal{G}_{E}\rdbrack.$

######  Proposition 5.13.

The groupoid $\mathcal{G}(G,E)$ obtained from the self-similar action in
Example 2.6 does not satisfy the property TR.

###### Proof.

Recall that from [3] it follows that $H_{1}(\mathcal{G}(G,E))\cong 0$, so the
kernel of the index map $I:\ldbrack\mathcal{G}(G,E)\rdbrack\to
H_{1}(\mathcal{G}(G,E))$ is the whole group
$\ldbrack\mathcal{G}(G,E)\rdbrack$. Consider the $G$-table

$\tau=\left(\begin{array}[]{cccc}u&v&e_{4}&e_{5}\\\ a^{-1}&cb&cb&a\\\
e_{4}&e_{5}&v&u\end{array}\right)$

which determines an element $\bar{\tau}\in\ldbrack\mathcal{G}(G,E)\rdbrack$.
Consider the transpositions

$\tau_{1}=\left(\begin{array}[]{cccc}u&e_{4}&e_{5}&v\\\ a^{-1}&a&v&v\\\
e_{4}&u&e_{5}&v\end{array}\right),\tau_{2}=\left(\begin{array}[]{cccc}v&e_{5}&e_{4}&u\\\
cb&b^{-1}c^{-1}&v&u\\\ e_{5}&v&e_{4}&u\end{array}\right),$
$\tau_{3}=\left(\begin{array}[]{ccc}u&v&w\\\ a^{-1}cb&b^{-1}c^{-1}a&w\\\
v&u&w\end{array}\right).$

A computation shows that

$\tau_{3}\tau_{2}\tau_{1}\tau=\left(\begin{array}[]{cccc}e_{4}&e_{5}&v&u\\\
v&v&v&(a^{-1}cba)^{2}\\\ e_{4}&e_{5}&v&u\end{array}\right),$

which is not a product of transpositions, since $G_{u}^{u}\cong\mathbb{Z}$. In
particular, $\tau$ is not a product of transpositions. ∎

######  Remark 5.14.

The groupoid $\mathcal{G}(G,E)$ obtained from the self-similar action in
Example 2.6 satisfies the HK conjecture, see [3]. The AH-conjecture may well
be true for this groupoid, but the technique of proof will be different from
the one in [19]. In particular, we would obtain that
$\ldbrack\mathcal{G}(G,E)\rdbrack_{ab}=0$ and that
$\ldbrack\mathcal{G}(G,E)\rdbrack$ is perfect.

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