# Hyperfiniteness and Borel asymptotic dimension of boundary actions of
hyperbolic groups

Petr Naryshkin  Petr Naryshkin, Mathematisches Institut, WWU Münster,
Einsteinstrasse 62, 48149 Münster, Germany<EMAIL_ADDRESS>and
Andrea Vaccaro  Andrea Vaccaro, Mathematisches Institut, WWU Münster,
Einsteinstrasse 62, 48149 Münster, Germany<EMAIL_ADDRESS>
###### Abstract.

We give a new short proof of the theorem due to Marquis and Sabok, which
states that the orbit equivalence relation induced by the action of a finitely
generated hyperbolic group on its Gromov boundary is hyperfinite. Our methods
permit moreover to show that every such action has finite Borel asymptotic
dimension.

The authors were funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2044 –
390685587, Mathematics Münster – Dynamics – Geometry – Structure, the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
427320536 – SFB 1442, and ERC Advanced Grant 834267 - AMAREC.

##  1\. Introduction

The study of countable Borel equivalence relations is a prominent subject in
descriptive set theory, whose interactions with dynamics have been thriving
ever since Feldman and Moore showed in [FM] that every countable Borel
equivalence relation is equal to the orbit equivalence relation of a Borel
action by a countable group. Due to this characterisation, the problem of
understanding which Borel actions generate hyperfinite equivalence relations
has attracted considerable attention in the past decades.

This line of research began with Weiss’ seminal work [weiss], where he proved
that any Borel action of ${\mathbb{Z}}$ gives rise to a hyperfinite orbit
equivalence relation, and asked whether this holds for all amenable groups.
His question was followed by more than 30 years of research during which
larger and larger classes of amenable groups have been shown to generate
hyperfinite equivalence relations ([JKL, GJ, SS]), culminating in the recent
paper [ConJacMarSewTuc20]. The main novelty of [ConJacMarSewTuc20] is the
notion of _Borel asymptotic dimension_ (Definition 4.1), a Borel analogue of
the homonymous large-scale invariant introduced by Gromov in [gromov], which
provides a sufficient condition to verify hyperfiniteness. More precisely, if
$X$ is a standard Borel space with extended metric $\rho$ (i.e. which also
allows value $\infty$) and $(X,\rho)$ has finite Borel asymptotic dimension,
then the finite-distance equivalence relation is hyperfinite
([ConJacMarSewTuc20, Theorem 1.7]). Thanks to this innovative approach, the
authors of [ConJacMarSewTuc20] were able to provide a positive answer to
Weiss’ question for the largest class of amenable groups to date.

Turning to non-amenable groups, Weiss’ question is known to have negative
answer. Indeed, every non-amenable group admits a Borel action whose orbit
equivalence relation is not hyperfinite ([JKL, Corollary 1.8]). Nevertheless,
it is still worthwhile to study specific natural actions of non-amenable
groups, particularly when the orbit equivalence relation is amenable. In this
direction, [DJK, Corollary 8.2] shows that the tail equivalence relation on
$S^{\mathbb{N}}$ is hyperfinite whenever $S$ is countable. An immediate
consequence of this is that the orbit equivalence relation of the action of a
finitely generated free group $\mathbb{F}$ on its Gromov boundary
$\partial\mathbb{F}$ is hyperfinite. Indeed, $\partial\mathbb{F}$ naturally
embeds into $S^{\mathbb{N}}$, where $S$ is a symmetric set of free generators,
and with this identification the orbit equivalence relation on
$\partial\mathbb{F}$ coincides with the tail equivalence on $S^{\mathbb{N}}$.

Building on this result, in [cubulated] it is proved that the orbit
equivalence relation of the boundary action $G\curvearrowright\partial G$ is
hyperfinite for all cubulated hyperbolic groups $G$, and this was later
successfully extended in [MarSab] to all finitely generated hyperbolic groups.
More recently, a further generalization of these methods was carried in
[relative], where a similar theorem is obtained for relatively hyperbolic
groups. More results on hyperfiniteness of orbit equivalence relations of
boundary actions can be found in [PrSa].

The key difference between a general hyperbolic group, equipped with a finite
generating set $S$, and a free group is that for the former there is no
canonical embedding $\partial G\hookrightarrow S^{\mathbb{N}}$ that maps the
orbit equivalence to a well-understood equivalence relation of the image. To
fill this gap, the papers [cubulated, MarSab, relative] develop sophisticated
geometric techniques to study families of geodesic ray bundles, and to adapt
the methods used in [DJK] to general boundary actions of hyperbolic groups.

The first main contribution of this paper is a new short proof of the main
result of [MarSab], stated explicitly below, which avoids most of the
geometric machinery employed in [MarSab] and provides a direct connection
between the orbit equivalence relation of $G\curvearrowright\partial G$ and
the tail equivalence on $S^{\mathbb{N}}$.

######  Theorem A ([MarSab, Theorem A]).

The orbit equivalence relation induced by the action of a finitely generated
hyperbolic group on its Gromov boundary is hyperfinite.

Our strategy to prove Theorem A starts with fixing some linear order on a
finite generating set $S$ of $G$. This induces a well-defined Borel function
mapping each $\eta\in\partial G$ to the unique geodesic ray converging to it
which is minimal (or rather whose _type_ is minimal, see §3) in the
lexicographical order on $S^{\mathbb{N}}$. That provides an embedding
$\Phi\colon\partial G\hookrightarrow S^{\mathbb{N}}$ (Definition 3.2). While
it might not be the case that the orbit equivalence is mapped to the tail
equivalence under this map, we show in Lemma 4.3 that, thanks to
hyperbolicity, the former contains only finitely many classes of the latter.
The result then follows from [JKL, Proposition 1.3 (vii)].

Another advantage of this approach is that our methods can be easily adapted
to the framework introduced in [ConJacMarSewTuc20]. Indeed, if $\rho_{s}$ is
the extended metric on $S^{\mathbb{N}}$ induced by the left shift $s\colon
S^{\mathbb{N}}\to S^{\mathbb{N}}$, then [ConJacMarSewTuc20, Lemma 8.3] shows
that $(S^{\mathbb{N}},\rho_{s})$ has Borel asymptotic dimension 1. Our
technical result Lemma 4.3 shows that we can essentially pull back this
information via the aforementioned embedding $\Phi\colon\partial G\to
S^{\mathbb{N}}$ to $\partial G$, and obtain the following result.

######  Theorem B.

The action of a finitely generated hyperbolic group on its Gromov boundary has
finite Borel asymptotic dimension.

The paper is organized as follows. In §2 we recall some preliminaries on
hyperbolic groups and prove a few basic facts about geodesic rays. In §3 we
isolate a specific Borel uniformizing function from $\partial G$ to
$S^{\mathbb{N}}$. In §4 we recall the definitions of hyperfiniteness and Borel
asymptotic dimension and give the proofs of Theorems A and B.

##  2\. Preliminaries on hyperboilc groups

We start with some essential preliminaries and notation on graphs, geodesic
rays and hyperbolicity. We refer to [MarSab]*§2 for a more detailed, self-
contained introduction on these topics (for a more throughout presentation of
this subject we recommend instead [BH]).

A _graph_ $X=(V,{\mathcal{E}})$ is a pair composed of a set of _vertices_ $V$
and a (symmetric) set of _edges_ ${\mathcal{E}}\subseteq V\times V$. Two
vertices $x,y\in V$ are _adjacent_ if $(x,y)\in{\mathcal{E}}$, and a _path_
$\Gamma$ in $X$ is a (possibly infinite) sequence of vertices $(x_{n})_{n\in
I}$ indexed by an interval $I\subseteq{\mathbb{N}}$ such that
$(x_{n},x_{n+1})\in{\mathcal{E}}$ for every $n\in I$. We shall sometimes
denote $x_{n}$ by $\Gamma(n)$. The _graph metric_ on $V$ is defined by setting
$d(x,y)$ as the smallest possible length of a path joining $x$ and $y$, and it
has value $\infty$ if no such path exists. A _geodesic path_ is a path
$(x_{n})_{n\in I}$ such that $d(x_{n},x_{m})=|n-m|$ for every $n,m\in I$. Two
infinite geodesic paths $\Gamma_{0},\Gamma_{1}$ are _asymptotic_ if their
Hausdorff distance $d_{H}(\Gamma_{0},\Gamma_{1})$ is finite. The _visual
boundary_ $\partial X$ of the graph $X$ is the set of equivalence classes of
asymptotic infinite geodesic paths. Given $x\in V$ and $\eta\in\partial X$, a
_combinatorial geodesic ray_ (or simply a _CGR_) from $x$ to $\eta$ is a
geodesic path $\Gamma=(x_{n})_{n\in{\mathbb{N}}}$ whose equivalence class in
$\partial X$ is $\eta$ and such that $x_{0}=x$. In this case, we say that
$\Gamma$ _starts_ at $x_{0}$ and _converges_ to $\eta$. Let
${\mathrm{CGR}}(x,\eta)$ denote the set of all CGR starting from $x$ and
converging to $\eta$.

Given $\delta>0$, a locally finite graph $X=(V,{\mathcal{E}})$ (i.e. such that
every vertex has finitely many adjacent vertices) is _$\delta$ -hyperbolic_ if
for every geodesic triangle (a triangle whose sides are geodesic paths)
$\Delta$ of $X$, each side of $\Delta$ is contained in the
$\delta$-neighborhood of the other two sides. In this case $\partial X$ can be
equipped with a compact metrizable topology as follows. Fix a base point $x\in
V$. We say that a sequence $\eta_{n}\in\partial X$ converges to some
$\eta\in\partial X$ if and only if there are
$\Gamma_{n}\in{\mathrm{CGR}}(x,\eta_{n})$ such that every subsequence of
$(\Gamma_{n})_{n\in{\mathbb{N}}}$ has a subsequence converging to some
$\Gamma\in{\mathrm{CGR}}(x,\eta)$ in the space $V^{\mathbb{N}}$ equipped with
the product topology resulting from the discrete topology on $V$. The set
$\partial X$ equipped with such topology, which does not depend on the choice
of the base point $x$, is called the _Gromov boundary_ of $X$ (see
[BH]*§III.3).

Let $G$ be a finitely generated group and let $S$ be a finite symmetric
generating set of $G$. The _Cayley graph_ $\mathrm{Cay}(G,S)$ is the locally
finite graph whose vertex set is $G$ and where $g,h\in G$ are adjacent if and
only if $g^{-1}h\in S$. The group $G$ is _hyperbolic_ is there exists
$\delta>0$ such that $\mathrm{Cay}(G,S)$ is $\delta$-hyperbolic. It is well-
known that being hyperbolic is invariant with respect of quasi-isometry, and
therefore it does not depend on the choice of $S$. We denote
$\partial\mathrm{Cay}(G,S)$ simply as $\partial G$, since also its topology
does not depend on $S$ (see [BH]*Theorem 3.9). The action of $G$ on its Cayley
graph naturally induces an action on combinatorial geodesic rays which
preserves asymptoticity. This in turn results in a continuous amenable action
$G\curvearrowright\partial G$ on its Gromov boundary (see e.g. [ADR, Theorem
B.3.4]).

We isolate a couple of elementary properties of hyperbolic graphs which will
play an important role in our arguments in §4.

######  Lemma 2.1 ([BH]*Lemma III.3.3).

Let $X$ be a $\delta$-hyperbolic graph and let $\eta\in\partial X$. Suppose
that $\Gamma_{0}=\left(x_{n}\right)_{n\in{\mathbb{N}}}$ and
$\Gamma_{1}=\left(y_{n}\right)_{n\in{\mathbb{N}}}$ are CGR converging to
$\eta$.

  1. (1)

If $x_{0}=y_{0}$ then

$d(x_{n},y_{n})\leq 2\delta,\text{ for every }n\in{\mathbb{N}}.$

  2. (2)

In general, there are

$0\leq T_{0},T_{1}\leq 2(d(x_{0},y_{0})+3\delta+2)$

such that

$d(x_{T_{0}+n},y_{T_{1}+n})\leq 5\delta,\text{ for every }n\in{\mathbb{N}}.$

###### Proof.

Item (1) is exactly [BH]*Lemma III.3.3.(1).

For item (2), the existence of $T_{0},T_{1}\geq 0$ such that
$d(x_{T_{0}+n},y_{T_{1}+n})\leq 5\delta$ for every $n\in{\mathbb{N}}$ is
granted by [BH]*Lemma III.3.3.(2). The upper bound $T_{0},T_{1}\leq
2(d(x_{0},y_{0})+3\delta+2)$ does not appear explicitly in the lemma, but can
be deduced from the proof therein.

More precisely, let $c_{n}$ be a sequence of geodesic paths from $x_{0}$ to
$y_{n}$. As $X$ is locally finite, there exists a subsequence of
$(c_{n})_{n\in{\mathbb{N}}}$ converging to some CGR $\Gamma$ whose starting
point is $x_{0}$. Given $n\in{\mathbb{N}}$, consider the geodesic triangle
whose edges are $c_{0}$, $c_{n}$ and the initial segment $(y_{0},\dots,y_{n})$
of $\Gamma_{1}$. By $\delta$-hyperbolicity, the path $c_{n}$ is contained in a
$\delta$-neighborhood of $c_{0}\cup\Gamma_{1}$. We claim that the vertex
$c_{n}(m)$ belongs to a $\delta$-neighborhood of $\Gamma_{1}$ for every $m\geq
d(x_{0},y_{0})+\delta+1$ and every $n$ for which $c_{n}(m)$ is defined.
Indeed, if this were not the case, then $c_{n}(m)$ would belong to a
$\delta$-neighborhood of $c_{0}$, hence giving

$d(c_{n}(m),x_{0})\leq d(x_{0},y_{0})+\delta<m,$

and contradicting the fact that $c_{n}$ is a geodesic path from $x_{0}$ to
$c_{n}(m)$. This in particular implies that $\Gamma$ converges to $\eta$. A
similar argument shows that if $R_{0}=d(x_{0},y_{0})+\lceil\delta\rceil+1$ and
$m\geq 2(d(x_{0},y_{0})+\delta+1)$, then $d(c_{n}(R_{0}),y_{m})>\delta$.

Summarizing, this means that there exists $R_{1}\leq
2(d(x_{0},y_{0})+\delta+1)$ such that $d(\Gamma(R_{0}),y_{R_{1}})\leq\delta$.
Notice that, given $n\in{\mathbb{N}}$, the vertex $\Gamma(R_{0}+n)$ could be
in the $\delta$-neighborhood of a $y_{\ell}$ for $\ell<R_{1}$, but by the
triangle inequality there exists a minimum $k_{0}\in{\mathbb{N}}$ such that
for every $n\geq k_{0}$ there is $n^{\prime}\in{\mathbb{N}}$ satisfying

$d(\Gamma(R_{0}+n),y_{R_{1}+n^{\prime}})\leq\delta.$

Again by the triangle inequality it follows that $|n-n^{\prime}|\leq 2\delta$,
hence

$d(\Gamma(R_{0}+n),y_{R_{1}+n})\leq 3\delta\text{ for every }n\geq k_{0}.$

Since both $\Gamma_{0}$ and $\Gamma$ belong to ${\mathrm{CGR}}(x_{0},\eta)$,
then $d(\Gamma(n),x_{n})\leq 2\delta$ for every $n\in{\mathbb{N}}$ by part
(1), thus we get

$d(x_{R_{0}+k_{0}+n},y_{R_{1}+k_{0}+n})\leq 5\delta\text{ for every
}n\in{\mathbb{N}}.$

We claim that $T_{i}=R_{i}+k_{0}$ satisfy the required bound. If $k_{0}=0$
there is nothing to prove since both $R_{0}$ and $R_{1}$ are smaller than
$2(d(x_{0},y_{0})+\delta+1)$. If instead $k_{0}>0$, let $m\in{\mathbb{N}}$ be
such that

$d(\Gamma(R_{0}+k_{0}),y_{R_{1}+m})\leq\delta.$

The triangle inequality implies $|k_{0}-m|\leq 2\delta$ and, as
$\Gamma(R_{0}+k_{0}-1)$ is in the $\delta$-neighborhood of some $y_{s}$ for
$s<R_{1}$, it also implies $m\leq 2\delta+1$. We can thus conclude that
$k_{0}\leq 4\delta+1$, and that $T_{0}$ and $T_{1}$ satisfy the required upper
bound also when $k_{0}>0$. ∎

Given a finite path $\Gamma_{0}=(x_{n})_{n<k}$ and a path
$\Gamma_{1}=(y_{n})_{n\in I}$ in a graph $X=(V,\mathcal{E})$ such that
$(x_{k-1},y_{0})\in\mathcal{E}$, we denote by $\Gamma_{0}\cdot\Gamma_{1}$ the
path which starts with $\Gamma_{0}$ and then continues along $\Gamma_{1}$.

######  Lemma 2.2.

Let $X$ be a graph, let $\Gamma=\left(x_{n}\right)_{n\in{\mathbb{N}}}$ be a
CGR, let $n_{1}>n_{0}\in{\mathbb{N}}$ and let $\Gamma_{01}$ be any geodesic
path from $x_{n_{0}}$ to $x_{n_{1}}$. Then
$\left(x_{n}\right)_{n<n_{0}}\cdot\Gamma_{01}\cdot\left(x_{n}\right)_{n>n_{1}}$
is a CGR.

###### Proof.

By assumption $\Gamma$ is a CGR, hence $d(x_{0},x_{n-1})=n-1$ for every
$n\in{\mathbb{N}}$. Let
$\left(y_{n}\right)_{n\in{\mathbb{N}}}=\left(x_{n}\right)_{n<n_{1}}\cdot\Gamma_{01}\cdot\left(x_{n}\right)_{n>n_{2}}$
and suppose that it is not a CGR. Then there exists $N\in{\mathbb{N}}$ such
that for all $n>N$ we have $d(y_{0},y_{n-1})<n-1$. However, $y_{0}=x_{0}$ and
$\Gamma_{01}$ is a geodesic path between $x_{n_{0}}$ and $x_{n_{1}}$, hence
$y_{n}=x_{n}$ for every $n>n_{1}$. This contradicts the fact that
$\left(x_{n}\right)_{n\in{\mathbb{N}}}$ is a CGR. ∎

##  3\. Borel Uniformizing Functions

Given a hyperbolic group $G$ with finite symmetric generating set $S$, in this
section we define a specific Borel injection which associates to each
$\eta\in\partial G$ a sequence in $S^{\mathbb{N}}$. Let
$\Gamma=(x_{n})_{n\in{\mathbb{N}}}\in G^{\mathbb{N}}$ be a CGR in
$\text{Cay}(G,S)$. Its _type_ $\text{typ}(\Gamma)$ is defined as

$\text{typ}(\Gamma)=(x_{n}^{-1}x_{n+1})_{n\in{\mathbb{N}}}\in S^{\mathbb{N}}.$

We use an analogous notation also for finite paths.

In what follows we consider $\partial G$ with its topology as Gromov boundary,
while $S^{\mathbb{N}}$ and $G^{\mathbb{N}}$ are endowed with the product
topology resulting from the discrete topology on $S$ and $G$ respectively.

######  Proposition 3.1 ([MarSab]*Lemma 5.1).

The set ${\mathrm{CGR}}(e,\eta)$ is compact in $G^{\mathbb{N}}$.

###### Proof.

As $\text{Cay}(G,S)$ is locally finite, any sequence
$\left(\Gamma_{n}\right)_{n\in{\mathbb{N}}}\subseteq{\mathrm{CGR}}(e,\eta)$
has a subsequence converging to some geodesic ray $\Gamma$. It follows from
Lemma 2.1 that $d(\Gamma(m),\Gamma_{0}(m))<2\delta$ for all $m\in{\mathbb{N}}$
which shows that $\Gamma$ converges to $\eta$. ∎

######  Definition 3.2.

Let $G$ be a hyperbolic group with finite symmetric generating set $S$. Fix a
total order on $S$ and let $\leq_{\mathrm{lex}}$ be the resulting
lexicographical order on $S^{\mathbb{N}}$. Consider the map

$\displaystyle\Phi\colon\partial G$ $\displaystyle\to S^{\mathbb{N}}$
$\displaystyle\eta$ $\displaystyle\mapsto\sigma_{\eta}$

where $\sigma_{\eta}$ is the unique element in $S^{\mathbb{N}}$ such that

$\sigma_{\eta}=\min\nolimits_{\leq_{\mathrm{lex}}}\\{\text{typ}(\Gamma)\mid\Gamma\in{\mathrm{CGR}}(e,\eta)\\},$

with $e$ being the identity element in $G$. The existence of such minimal
element follows from Proposition 3.1.

######  Proposition 3.3.

The function $\Phi\colon\partial G\to S^{\mathbb{N}}$ in Definition 3.2 is a
Borel injection.

###### Proof.

Consider the set

$A=\\{(\eta,\Gamma)\in\partial G\times
G^{\mathbb{N}}\mid\Gamma\in\text{CGR}(e,\eta)\\}.$

The proof of [MarSab]*Claim 6.3 shows that $A$ is closed in $\partial G\times
G^{\mathbb{N}}$. Moreover, given $\eta\in\partial G$, the section

$A_{\eta}=\\{\Gamma\in G^{\mathbb{N}}\mid(\eta,\Gamma)\in A\\}$

is precisely ${\mathrm{CGR}}(e,\eta)$ which is a compact subset of
$G^{\mathbb{N}}$ by Proposition 3.1. Then by [Kech]*Theorem 28.8 the map

$\displaystyle\Theta\colon\partial G$ $\displaystyle\to K(G^{\mathbb{N}})$
$\displaystyle\eta$ $\displaystyle\mapsto{\mathrm{CGR}}(e,\eta)$

is Borel, where $K(G^{\mathbb{N}})$ is the space of compact subsets if
$G^{\mathbb{N}}$ with the Vietoris topology (see [Kech, §I.4.F]). Consider
next the function

$\displaystyle\Psi\colon K(G^{\mathbb{N}})$ $\displaystyle\to S^{\mathbb{N}}$
$\displaystyle F$
$\displaystyle\mapsto\min\nolimits_{\leq_{\mathrm{lex}}}\\{\text{typ}(\Gamma)\mid\Gamma\in
F\\}.$

We claim that also $\Psi$ is Borel. Indeed, let $U$ be the basic open set in
$S^{\mathbb{N}}$ consisting of all sequences starting with some
$(s_{0},\dots,s_{k-1})\in S^{k}$. Let $U^{\prime}$ be the open set in
$G^{\mathbb{N}}$ of all elements starting with $(x_{0},\dots,x_{k-1})$, where
$x_{n}=s_{0}s_{1}\dots s_{n}$ for every $n<k$. Given
$\bar{y}=(y_{0},\dots,y_{k-1})\in G^{k}$ denote the corresponding basic open
set in $G^{\mathbb{N}}$ as

$V_{\bar{y}}=\\{(z_{n})_{n\in{\mathbb{N}}}\in G^{\mathbb{N}}\mid
z_{n}=y_{n}\text{ for all }n<k\\}.$

Let $C=\\{\bar{y}\in
G^{k}\mid(s_{0},\dots,s_{k-1})\leq_{\mathrm{lex}}\text{typ}(\bar{y})\\}$ and
consider the open set

$V=\bigcup_{\bar{y}\in C}V_{\bar{y}}.$

The preimage $\Psi^{-1}(U)$ consists of all $F\in K(G^{\mathbb{N}})$ which
intersect $U^{\prime}$ and which are contained in $V$, and it is therefore and
open set in the Vietoris topology of $K(G^{\mathbb{N}})$. We can thus conclude
that $\Phi=\Psi\circ\Theta$ is Borel.

Injectivity of $\Phi$ follows since any two CGR which start from the same
point and have the same type must be equal. ∎

##  4\. Hyperfiniteness and Borel Asymptotic Dimension

This section is devoted to the proof of Theorem 4.2, which covers both Theorem
A and B. We begin with some preliminaries on Borel equivalence relations and
by recalling the notion of Borel asymptotic dimension as introduced in
[ConJacMarSewTuc20].

An _extended metric_ $\rho$ on a set $X$ is a metric on $X$ which can also
take value $\infty$. For $x\in X$ and $r>0$, let $B_{\rho}(x,r)$ denote the
$\rho$-ball of radius $r$ centered in $x$. Let $E_{\rho}$ be the equivalence
relation of finite distance between points

$E_{\rho}=\\{(x,y)\in X\times X\mid\rho(x,y)<\infty\\}.$

Given a standard Borel space $X$, an equivalence relation $E$ on $X$ is
_Borel_ if $E$ is a Borel subset of $X\times X$. A Borel equivalence relation
$E$ is _finite_ (respectively _countable_) if all its equivalence classes are
finite (respectively countable), and it is _hyperfinite_ if it can be written
as a union of an increasing sequence of finite Borel equivalence relations.

A _Borel extended metric space_ $(X,\rho)$ is a standard Borel space paired
with a Borel extended metric $\rho$. In this case $E_{\rho}$ is a Borel
equivalence relation on $X$. An equivalence relation $E$ on a Borel extended
metric space $(X,\rho)$ is _uniformly bounded_ if there exists a uniform
finite bound on the $\rho$-diameter of the $E$-classes. A graph
$X=(V,{\mathcal{E}})$ is a _Borel graph_ if $V$ is a standard Borel space and
the edge relation ${\mathcal{E}}$ is a Borel subset of $V\times V$. Note that
the graph metric endows $V$ with a structure of Borel extended metric space.

We will mainly be concerned with two types of Borel graphs. The first one is
the _Schreier graph_ of a Borel action of a finitely generated group $G$ on a
standard Borel space $X$. Given a finite symmetric generating set $S$ of $G$,
the Schreier graph of the action $G\curvearrowright X$ has $X$ as vertex set,
and two points $x,y\in X$ are adjacent if and only if there is $g\in S$ such
that $gx=y$. We remark that all vertices of such graph have degree at most
$|S|$, hence for every $r>0$ all balls of radius $r$ (in the graph metric)
have size at most $|S|^{r}$.

The _orbit equivalence relation_ $E_{G}^{X}$ on $X$ is defined as

$xE_{G}^{X}y\iff\exists g\in G\text{ s.t. }gx=y,\text{ for }x,y\in X.$

Note that the orbit equivalence relation is the same as the finite-distance
equivalence relation induced by the graph metric on $X$.

The second kind of Borel graphs we focus on arises from Borel functions. Let
$X$ be a standard Borel space and let $f\colon X\to X$ be a Borel function. We
let $\mathcal{G}_{f}$ be the Borel graph whose vertex set is $X$, and where
$x,y\in X$ are adjacent if and only if $f(x)=y$ or $f(y)=x$. We denote the
graph metric on $\mathcal{G}_{f}$ by $\rho_{f}$. The typical example we shall
consider is the left shift on $X=S^{\mathbb{N}}$ for a finite set $S$

(4.1) $\displaystyle s\colon S^{\mathbb{N}}$ $\displaystyle\to S^{\mathbb{N}}$
$\displaystyle(x_{n})_{n\in{\mathbb{N}}}$
$\displaystyle\mapsto(x_{n+1})_{n\in{\mathbb{N}}}$

In this case the finite-distance equivalence relation $E_{\rho_{s}}$
corresponds to the _tail equivalence_ $R$ on sequences in $S$, that is for
$\sigma_{0}=(x_{n})_{n\in{\mathbb{N}}},\sigma_{1}=(y_{n})_{n\in{\mathbb{N}}}\in
S^{\mathbb{N}}$

$\sigma_{0}R\sigma_{1}\iff\exists T_{0},T_{1}\in{\mathbb{N}}\text{ s.t.
}x_{T_{0}+n}=y_{T_{1}+n},\text{ for every }n\in{\mathbb{N}}.$

Finally, we recall the notion of Borel asymptotic dimension.

######  Definition 4.1 ([ConJacMarSewTuc20, Definition 3.2]).

Let $(X,\rho)$ be a Borel extended metric space such that $E_{\rho}$ is
countable. The _Borel asymptotic dimension_ $\text{asdim}_{B}(X,\rho)$ of $X$
is the smallest $d\in{\mathbb{N}}$ such that for every $r>0$ there exists a
uniformly bounded Borel equivalence relation $E$ on $X$ such that for every
$x\in X$ the ball $B_{\rho}(x,r)$ intersects at most $d+1$ classes of $E$, and
it is $\infty$ if no such $d$ exists.

If $G\curvearrowright X$ is a Borel action, then we denote the Borel
asymptotic dimension of the corresponding Schreier graph simply as
$\text{asdim}_{B}(G\curvearrowright X)$ (note that it does not depend on the
choice of the generating set $S$ of $G$ inducing the metric on $X$ by
[ConJacMarSewTuc20, Lemma 2.2]).

The main goal of this section is proving the following theorem.

######  Theorem 4.2.

Let $G$ be a hyperbolic group with finite symmetric generating set $S$.

  1. (1)

The orbit equivalence relation $E^{\partial G}_{G}$ induced by the boundary
action $G\curvearrowright\partial G$ is hyperfinite.

  2. (2)

Let $\delta>0$ be such that $\text{Cay}(G,S)$ is $\delta$-hyperbolic and let
$M_{5\delta}\in{\mathbb{N}}$ be a bound on the size of all balls of radius
$5\delta$ in $\text{Cay}(G,S)$. Then
$\text{asdim}_{B}(G\curvearrowright\partial G)\leq 2M_{5\delta}^{2}-1$.

The main motivation behind Definition 4.1 is [ConJacMarSewTuc20, Theorem 1.7],
stating that if a Borel extended metric space $(X,\rho)$ has finite Borel
asymptotic dimension, then the equivalence relation $E_{\rho}$ is hyperfinite.
In view of this, it would be enough to prove item (2) to also obtain part (1).
We point out, however, that a small modification of the methods employed to
prove the second statement of Theorem 4.2 can be used to show that
$E_{G}^{\partial G}$ is hyperfinite, without having to resort to the theory
developed in [ConJacMarSewTuc20]. We will include this more direct proof of
the first part of Theorem 4.2 for the reader’s convenience.

Let us first consider Theorem 4.2 in the case where $G=\mathbb{F}_{2}$, the
free group with two generators $a,b$. Let $S=\\{a,b,a^{-1},b^{-1}\\}$ and let
$s\colon S^{\mathbb{N}}\to S^{\mathbb{N}}$ be the left shift on
$S^{\mathbb{N}}$. Each $\eta\in\partial\mathbb{F}_{2}$ has a unique
representative in ${\mathrm{CGR}}(e,\eta)$, and this gives a Borel injection
$\Phi:\eta\mapsto\sigma_{\eta}$ from $\partial\mathbb{F}_{2}$ to
$S^{\mathbb{N}}$ such that

(4.2) $\text{either}\ \sigma_{g\eta}=s(\sigma_{\eta})\ \text{or}\
\sigma_{\eta}=s(\sigma_{g\eta})\text{ for every
}\eta\in\partial\mathbb{F}_{2}\text{ and }g\in S.$

It follows that $\Phi$ maps the Shreier graph metric on
$\partial\mathbb{F}_{2}$ to the metric $\rho_{s}$ on $S^{\mathbb{N}}$. Thanks
to this, the analysis of the orbit equivalence relation on
$\partial\mathbb{F}_{2}$ is equivalent to that of the tail equivalence
relation on $S^{\mathbb{N}}$, hence item (1) of Theorem 4.2 directly follows
from [DJK, Corollary 8.2] and item (2) from [ConJacMarSewTuc20, Lemma 8.3].
Our strategy to prove Theorem 4.2 is to use the Borel injection $\Phi$ from
Definition 3.2 to have a similar reduction to the tail equivalence relation on
$S^{\mathbb{N}}$. The situation is however more chaotic for a general
hyperbolic group, since we cannot expect for an equality like the one in (4.2)
to hold. Nevertheless, hyperbolicity grants a sufficient control of the tail
equivalence classes of $\sigma_{g\eta}$, as $g$ varies in a given bounded ball
around the identity of $G$, to fill this gap.

This will be made precise in the following lemma, for which we introduce some
notation. Let $G$ be a hyperbolic group with finite symmetric generating set
$S$, let $\delta>0$ be such that $\text{Cay}(G,S)$ is $\delta$-hyperbolic and
let $M_{5\delta}\in{\mathbb{N}}$ be an upper bound for the cardinality of the
balls of radius $5\delta$ in $\text{Cay}(G,S)$.

######  Lemma 4.3.

Let $G$ be a hyperbolic group with finite symmetric generating set $S$. Fix
$\eta\in\partial G$, $r>0$, let $\rho$ be the graph metric on $\partial G$
induced by the action $G\curvearrowright\partial G$, and let $\rho_{s}$ be the
extended metric on $S^{\mathbb{N}}$ induced by the left shift $s\colon
S^{\mathbb{N}}\to S^{\mathbb{N}}$. Finally, let
$\\{\sigma_{\zeta}\\}_{\zeta\in B_{\rho}(\eta,r)}\subseteq S^{\mathbb{N}}$ be
the family of types of CGR given in Definition 3.2. Then
$\\{\sigma_{\zeta}\\}_{\zeta\in B_{\rho}(\eta,r)}$ can be partitioned into at
most $(M_{5\delta})^{2}$ sets, each one with $\rho_{s}$-diameter smaller than
$8(r+3\delta+2)$. In particular, the set $\\{\sigma_{\zeta}\\}_{\zeta\in
B_{\rho}(\eta,r)}$ meets at most $(M_{5\delta})^{2}$ tail equivalence classes
of $S^{\mathbb{N}}$.

###### Proof.

Given $\zeta\in\partial G$, let $\Gamma_{\zeta}\in{\mathrm{CGR}}(e,\zeta)$ be
the unique CGR starting from the identity $e\in G$ such that
$\text{typ}(\Gamma_{\zeta})=\sigma_{\zeta}$. Fix $\zeta\in B_{\rho}(\eta,r)$
and let $g_{\zeta}\in G$ be such that $d(e,g_{\zeta})<r$ and
$g_{\zeta}\zeta=\eta$. Set $\Gamma_{\eta}=(x_{n})_{n\in{\mathbb{N}}}$,
$\Gamma_{\zeta}=(z_{n})_{n\in{\mathbb{N}}}$ and let
$g_{\zeta}\Gamma_{\zeta}=(y_{n})_{n\in{\mathbb{N}}}\in G^{\mathbb{N}}$ be the
sequence where $y_{n}=g_{\zeta}z_{n}$ for every $n\in{\mathbb{N}}$. Then
$g_{\zeta}\Gamma_{\zeta}\in{\mathrm{CGR}}(g_{\zeta},\eta)$.

Set $T=2(r+3\delta+2)$. By Lemma 2.1, there is $0\leq T_{\zeta}\leq 2T$ such
that

$d\left(x_{T+n},y_{T_{\zeta}+n}\right)\leq 5\delta,\text{ for every
}n\in{\mathbb{N}}.$

Note that $g_{\zeta}\Gamma_{\zeta}$ is the unique ray in
${\mathrm{CGR}}(g_{\zeta},\eta)$ such that

(4.3)
$\text{typ}(g_{\zeta}\Gamma_{\zeta})=\min\nolimits_{\leq_{\mathrm{lex}}}\\{\text{typ}(\Gamma)\mid\Gamma\in{\mathrm{CGR}}(g_{\zeta},\eta)\\},$

since $\sigma_{\zeta}$ is the $\leq_{\mathrm{lex}}$-minimum of
$\text{typ}({\mathrm{CGR}}(e,\zeta))$ and shifting by $g_{\zeta}$ gives a
bijection between ${\mathrm{CGR}}(e,\zeta)$ and
${\mathrm{CGR}}(g_{\zeta},\eta)$ that preserves the type.

Fix $N\in{\mathbb{N}}$. By Lemma 2.2 and (4.3) the path
$p=\left(y_{T_{\zeta}+n}\right)_{n=0}^{N}$ is the unique geodesic between
$y_{T_{\zeta}}$ and $y_{T_{\zeta}+N}$ such that

$\text{typ}(p)=\min\nolimits_{\leq_{\mathrm{lex}}}\\{q\in S^{N}\mid
q=\text{typ}(r)\text{ for some geodesic path }r\text{ from
}y_{T_{\zeta}}\text{ to }y_{T_{\zeta}+N}\\}.$

Summarizing, the partial segment of $g_{\zeta}\Gamma_{\zeta}$ between the
coordinates $T_{\zeta}$ to $T_{\zeta}+N$ only depends on the initial point
$y_{T_{\zeta}}\in B_{d}(x_{T},5\delta)$ and on $y_{T_{\zeta}+N}\in
B_{d}(x_{T+N},5\delta)$. Since this is true for every $N\in{\mathbb{N}}$,
there are at most $(M_{5\delta})^{2}$ choices for
$\left(y_{T_{\zeta}+n}\right)_{n\in{\mathbb{N}}}$ as $\zeta$ varies in
$B_{\rho}(\eta,r)$. Since
$\text{typ}(g\Gamma_{\zeta})=\text{typ}(\Gamma_{\zeta})=\sigma_{\zeta}$ for
every $\zeta\in\partial G$ and every $g\in G$, the conclusion of the lemma
follows. ∎

We are now ready to prove the main result.

###### Proof of Theorem 4.2.

To prove (1), let $s$ be the left shift on $S^{\mathbb{N}}$ and let $R$ denote
the tail equivalence relation on $S^{\mathbb{N}}$. Consider the map
$\Phi\colon\partial G\to S^{\mathbb{N}}$ from Definition 3.2, and let $R_{0}$
be the preimage of $R$ on $\partial G$ under $\Phi$:

$\eta R_{0}\zeta\iff\sigma_{\eta}R\sigma_{\zeta},\text{ for every
}\eta,\zeta\in\partial G.$

Note that the equivalence relation $R$ is hyperfinite by [DJK, Corollary 8.2],
and thus $R_{0}$ is a Borel hyperfinite equivalence relation since $\Phi$ is a
Borel injection by Proposition 3.3. Let $R_{1}=R_{0}\cap E^{\partial G}_{G}$,
which by [JKL, Proposition 1.3.(i)] is again hyperfinite (this step is
actually redundant, as we show in Claim 4.4). Let $M_{5\delta}\in{\mathbb{N}}$
be an upper bound for the size of the balls of radius $5\delta$ in
$\text{Cay}(G,S)$. We claim that each $E^{\partial G}_{G}$-class is composed
of at most $(M_{5\delta})^{2}$ equivalence classes of $R_{1}$. This will grant
that $E^{\partial G}_{G}$ is hyperfinite by [JKL, Proposition 1.3.(vii)].

The claim is an immediate consequence of Lemma 4.3, since if
$\eta_{0},\dots,\eta_{(M_{5\delta})^{2}}$ are elements in $\partial G$ in the
same $E^{\partial G}_{G}$-class, then there exists $N>0$ such that
$\\{\eta_{k}\\}_{k\leq(M_{5\delta})^{2}}\subseteq B_{\rho}(\eta_{0},N)$, hence
there are $i<j\leq({M_{5\delta}})^{2}$ such that
$\sigma_{\eta_{i}}R\sigma_{\eta_{j}}$, and therefore $\eta_{i}R_{1}\eta_{j}$.

To prove (2), fix $r>0$ and set $T=8(r+3\delta+2)$. Consider $S^{\mathbb{N}}$
with the graph metric $\rho_{s}$ induced by the shift $s$. By
[ConJacMarSewTuc20, Lemma 8.3] the space $(S^{\mathbb{N}},\rho_{s})$ has Borel
asymptotic dimension at most 1. Hence there exists a uniformly bounded
equivalence relation $E$ such that for every $x\in S^{\mathbb{N}}$ the ball
$B_{\rho_{s}}(x,T)$, intersects at most 2 equivalence classes of $E$.

Let $E_{0}$ be the preimage of $E$ via $\Phi$ from Definition 3.2, that is for
$\eta,\zeta\in\partial G$ we define

$\eta E_{0}\zeta\iff\sigma_{\eta}E\sigma_{\zeta}.$

Let $\rho$ be the graph metric on $\partial G$ induced by the Schreier graph
of the boundary action $G\curvearrowright\partial G$. The equivalence relation
$E_{0}$ is Borel since $\Phi$ is Borel (Proposition 3.3) and it is uniformly
bounded since $E$ is and because of the following claim.

######  Claim 4.4.

Let $\eta,\zeta\in\partial G$ Then

$\rho(\eta,\zeta)\leq\rho_{s}(\sigma_{\eta},\sigma_{\zeta}).$

###### Proof.

Suppose that there is $M\in{\mathbb{N}}$ such that
$\rho_{s}(\sigma_{\eta},\sigma_{\zeta})\leq M$. There exist thus
$n_{0},n_{1}\in{\mathbb{N}}$ with $n_{0}+n_{1}\leq M$ and such that
$s^{n_{0}}(\sigma_{\eta})=s^{n_{1}}(\sigma_{\zeta})=\bar{s}$. Let
$s_{1},\dots,s_{n_{0}},t_{1},\dots,t_{n_{1}}\in S$ be such that

(4.4) $\sigma_{\eta}=(s_{1},\dots,s_{n_{0}})^{\smallfrown}\bar{s}\text{ and
}\sigma_{\zeta}=(t_{1},\dots,t_{n_{1}})^{\smallfrown}\bar{s}.$

Let $\Gamma_{\eta}\in{\mathrm{CGR}}(e,\eta)$ and
$\Gamma_{\zeta}\in{\mathrm{CGR}}(e,\zeta)$ be the unique CGR starting from $e$
such that $\text{typ}(\Gamma_{\eta})=\sigma_{\eta}$ and
$\text{typ}(\Gamma_{\zeta})=\sigma_{\zeta}$. Set $g=(s_{1}s_{2}\dots
s_{n_{0}})^{-1}$ and $h=(h_{1}h_{2}\dots h_{n_{1}})^{-1}$. Consider finally
the CGR $g\Gamma_{\eta}$, whose $n$-th entry is equal to $g\Gamma_{\eta}(n)$,
and $h\Gamma_{\zeta}$, whose $n$-th entry is equal to $h\Gamma_{\zeta}(n)$. It
follows from (4.4), and from the definition of $g$ and $h$, that
$g\Gamma_{\eta}$ and $h\Gamma_{\zeta}$ are tail equivalent in
$G^{\mathbb{N}}$, hence they converge to the same $\theta\in\partial G$. In
particular, this shows that $g\eta=h\zeta=\theta$, and therefore
$\rho(\eta,\zeta)\leq M$. ∎

We claim that, given $\eta\in\partial G$, the ball $B_{\rho}(\eta,r)$ meets at
most $2(M_{5\delta})^{2}$ classes of $E_{0}$. Take
$\eta_{0},\dots,\eta_{2(M_{5\delta})^{2}}\in B_{\rho}(\eta,r)$. By Lemma 4.3
there are $k_{0}<k_{1}<k_{2}\leq 2(M_{5\delta})^{2}$ such that
$\\{\sigma_{\eta_{k_{i}}}\\}_{i=0,1,2}$ are pairwise tail equivalent, and
moreover

$\rho_{s}(\sigma_{\eta_{k_{i}}},\sigma_{\eta_{k_{j}}})\leq T,\text{ for every
}i,j=0,1,2.$

By the choice of $E$, there are distinct $i,j\in\\{0,1,2\\}$ such that
$\sigma_{\eta_{k_{i}}}E\sigma_{\eta_{k_{j}}}$, and therefore
$\eta_{k_{i}}E_{0}\eta_{k_{j}}$. We can thus conclude that $B_{\rho}(\eta,r)$
intersects at most $2(M_{5\delta})^{2}$ classes of $E_{0}$. ∎

## References