# Boundary Representations of Locally Compact Hyperbolic Groups

Michael Glasner<EMAIL_ADDRESS>Faculty of Mathematics and
Computer Science, The Weizmann Institute of Science, 234 Herzl Street, Rehovot
7610001, ISRAEL

(Date: September 22, 2023)

###### Abstract.

We develop the theory of Patterson-Sullivan measures on the boundary of a
locally compact hyperbolic group, associating to certain left invariant
metrics on the group measures on the boundary. We later prove that for second
countable, non-elementary, unimodular locally compact hyperbolic groups the
associated Koopman representations are irreducible and their isomorphism type
classifies the metric on the group up to homothety and bounded additive
changes, generalizing a theorem of Garncarek on discrete hyperbolic groups. We
use this to answer a question of Caprace, Kalantar and Monod on type I
hyperbolic groups in the unimodular case.

###### Key words and phrases:

Locally compact hyperbolic groups, Boundary representations, Patterson
Sullivan measures, Type I groups, Double ergodicity.

###### 2020 Mathematics Subject Classification:

22D10,20F67,43A65,37A40

###### Contents

  1. 1 Introduction
    1. 1.1 Koopman and boundary representations
    2. 1.2 Type I hyperbolic groups
    3. 1.3 The space $\partial^{2}G$
    4. 1.4 Non unimodular groups
    5. 1.5 Structure of the paper
    6. 1.6 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Notation and conventions
    2. 2.2 Maps between metric spaces
    3. 2.3 Hyperbolic metric spaces
    4. 2.4 Locally compact hyperbolic groups
    5. 2.5 Non-singular actions and Koopman representations
  3. 3 Patterson-Sulivan Measures
  4. 4 Some Growth Estimates
  5. 5 Operators on $L^{2}(\mu)$
    1. 5.1 Estimates on $\pi(g)$
    2. 5.2 Kernel operators
  6. 6 Irreducibility of Boundary Representations
  7. 7 Rough Equivalence of Metrics
  8. 8 Type I Hyperbolic Groups
  9. 9 The Geodesic Flow
  10. 10 Double Ergodicity
  11. A Cocycles and Almost Cocycles
  12. B Actions Admitting a Borel Cross Section

##  1\. Introduction

###  1.1. Koopman and boundary representations

A locally compact group $G$ is called hyperbolic if for some (hence any)
compact generating set the coresponding word metric is Gromov hyperbolic.
Equivalently $G$ is hyperbolic if it admits a proper, cocompact, isometric
action on a proper, geodesic, Gromov hyperbolic metric space $X$. Examples
are:

  1. (1)

Rank one simple Lie groups with finite center.

  2. (2)

Minimal parabolic subgroups of rank one simple Lie groups with finite center.

  3. (3)

Groups acting properly and co-compactly on locally finite trees.

One of the most useful tools to study $G$ is the action of $G$ on its Gromov
boundary $\partial G$. Given a left invariant metric $d$ on $G$ which is quasi
isometric to a word metric and satisfies some mild assumptions we will
construct a measure $\mu$ on $\partial G$ called the Patterson-Sullivan
measure associated to $d$. The measure class $[\mu]$ is invariant and $\mu$ is
quasi conformal, i.e there exists $C\geq 1$ such that for all $g\in G$:

$\frac{1}{C}e^{-h(|g|-2(g,\xi))}\leq\frac{dg_{*}\mu}{d\mu}(\xi)\leq
Ce^{-h(|g|-2(g,\xi))}$

where $h$ is the critical exponent of $G$ with respect to $d$ and
$(\cdot,\cdot)$ is the Gromov product.

To such a nonsingular action one can associate a unitary representation of $G$
on $L^{2}(\mu)$ called the Koopman representation, defined by:

$[\pi(g)f](\xi)=\sqrt{\frac{dg_{*}\mu}{d\mu}(\xi)}f(g^{-1}\xi)$

The Koopman representation depends up to unitary equivalence only on the
measure class $[\mu]$.

Irreducibility of the Koopman representation implies ergodicity and can be
thought of as a mixing property of the action.

We will call the Koopman representations associated to the Patterson-Sullivan
measures boundary representations or Patterson-Sullivan representations.

Our first main theorem is the following:

######  Theorem 1.1.

Let $G$ be a second countable, unimodular, non-elementary locally compact
hyperbolic group. For any Borel measurable, left invariant metric $d$ on $G$
which is Gromov hyperbolic and quasi isometric to a word metric, the
associated Patterson-Sullivan representation $\pi$ is irreducible.

Our second main theorem provides a classification of the Patterson-Sullivan
representations in terms of the metric $d$.

######  Theorem 1.2.

Let $G$ be a second countable, unimodular, non-elementary locally compact
hyperbolic group. Given two Borel measurable, left invariant metrics
$d_{1},d_{2}$ on $G$ which are Gromov hyperbolic and quasi isometric to a word
metric, the corresponding Patterson-Sullivan representations $\pi_{1}$ and
$\pi_{2}$ are unitarily equivalent if and only if $d_{1}$ and $d_{2}$ are
roughly similar, that is, there exist constants $L,C>0$ such that for all
$g,h\in G$, $Ld_{1}(g,h)-C\leq d_{2}(g,h)\leq Ld_{1}(g,h)+C$. (Note this is
much stronger than a quasi-isometry since the multiplicative constant is the
same in the upper and lower bounds!)

These theorems generalize earlier works by Bader, Muchnik [3] and Garncarek
[15] which deal with the cases of fundamental groups of negatively curved
manifolds and discrete hyperbolic groups respectively. Many of our methods are
similar to those in [15] although some constructions are altered to fit the
locally compact setting. It is interesting to notice throughout the paper the
key lemmas in which we use the unimodularity assumption from theorems 1.1 and
1.2. Specifically note lemma 4.3, lemma 5.6 and appendix B.

Theorem 1.1 can be envisioned in the wider context of a general conjecture by
Bader and Muchnik:

######  Conjecture (Bader, Muchnik [3]).

Let $G$ be a locally compact group and $\nu$ a spread out probability measure
on $G$. For any $\nu$-boundary of $G$ the associated Koopman representation is
irreducible.

###  1.2. Type I hyperbolic groups

Our main application of theorems 1.1 and 1.2 will be to the study of type I
hyperbolic groups, generalizing results of Caprace, Kalantar and Monod from
[11].

######  Definition 1.3.

A locally compact group $G$ is of type I if any two irreducible unitary
representations of $G$ which are weakly equivalent (see [4, Appendix F.1]) are
unitarily equivalent.

Type I groups have a relatively simple unitary representation theory. In a
certain sense they are the groups for which there is hope of obtaining a
complete classification of all unitary representations. For an introduction on
type I groups see [14, Section 7.2].

In section 8 we use theorems 1.1 and 1.2 to deduce the following
generalization of [11, Theorem B]:

######  Theorem 1.4.

Let $G$ be a second countable unimodular locally compact hyperbolic group. If
$G$ is of type I then $G$ has a co-compact amenable subgroup.

By a theorem of Thoma [20] [21], discrete groups are of type I if and only if
they are virtually abelian. Therefore discrete hyperbolic groups of type I are
elementary. As a result the above theorem is interesting only for non discrete
groups.

In [10, Theorem D] there is a classification of non amenable hyperbolic
locally compact groups containing a cocompact amenable subgroup. We can use
this to deduce:

######  Corollary 1.5.

Let $G$ be a second countable unimodular locally compact hyperbolic group.
Recall $G$ contains a unique maximal compact normal subgroup $W$. If $G$ is of
type I then exactly one of the following holds:

  1. (1)

$G/W$ is a rank one adjoint Lie group.

  2. (2)

$G/W$ is a closed subgroup of the automorphism group of a locally finite, non
elementary tree $T$, acting without inversions, with exactly two orbits of
vertices and $2$-transitively on the boundary.

  3. (3)

$G/W$ is trivial or isomorphic to
$\mathbb{Z},\mathbb{R},\mathbb{Z}\rtimes\\{\pm 1\\}$ or
$\mathbb{R}\rtimes\\{\pm 1\\}$.

It is already pointed out in [11, Remark 5.6] that generalizing the works of
Garncarek [15] as in theorems 1.1 and 1.2 would imply the theorems above.

Theorem 1.4 fits in to a much wider structural conjecture of Caprace Kalantar
and Monod about type I groups:

######  Conjecture (Caprace, Kalantar, Monod [11]).

Every second countable locally compact group of type I admits a cocompact
amenable subgroup.

###  1.3. The space $\partial^{2}G$

Let $G$ be a second countable locally compact hyperbolic group, $d$ a Borel
measurable left invariant metric on $G$ which is quasi isometric to a word
metric and Gromov hyperbolic. Denote by $\mu$ the corresponding Patterson
Sullivan measure.

During the proof of theorem 1.2 we will establish several results of
independent interest about the space $\partial^{2}G$ of distinct pairs in
$\partial G$.

First we shall construct an invariant (infinite) measure $m$ in the measure
class of $\mu^{2}$. This measure will be the analogue of the Bowen-Margulis-
Sullivan measure in our case.

It will then be shown that if $G$ is unimodular, then the action of $G$ on
$(\partial^{2}G,m)$ is weakly mixing in the following sense:

######  Theorem 1.6.

For any ergodic p.m.p action $G\operatorname{\curvearrowright}(\Omega,\omega)$
the diagonal action of $G$ on $(\partial^{2}G\times\Omega,m\times\omega)$ is
ergodic.

This implies in particular that the action of $G$ on $(\partial^{2}G,m)$ is
ergodic and therefore that $m$ is the unique invariant measure in the measure
class of $[\mu^{2}]$. This will be the critical fact about $\partial^{2}G$
used in the proof of theorem 1.2.

In order to prove theorem 1.6 we define a cocycle
$\tau:G\times\partial^{2}G\to\mathbb{R}$:

$\tau(g,\xi,\eta)=\frac{1}{h}\left(ln\frac{dg^{-1}_{*}\mu}{d\mu}(\eta)-ln\frac{dg^{-1}_{*}\mu}{d\mu}(\xi)\right)$

where $h$ is the critical exponent of $(G,d)$.

$G$ then acts on $\partial^{2}G\times\mathbb{R}\times\Omega$ via:

$g(\xi,\eta,t,w)=(g\xi,g\eta,t+\tau(g,\xi,\eta),gw)$

and this action commutes with the $\mathbb{R}$-flow defined by:

$\Phi^{s}(\xi,\eta,t,w)=(\xi,\eta,t+s,w)$

This flow provides an analogue of the geodesic flow of a negatively curved
simply connected manifold in our general context. Since it commutes with the
$G$ action $\Phi$ descends to an $\mathbb{R}$-flow $\phi$ on the space
$X=(\partial^{2}G\times\mathbb{R}\times\Omega)//G$ of $G$-ergodic components.
We will show that $X$ supports a $\phi$-invariant probability measure $\nu$ in
its canonical measure class.

Using $\phi$ we will prove the following ergodic theorem for $\partial^{2}G$:

######  Theorem 1.7.

Suppose $G$ is unimodular and let
$G\operatorname{\curvearrowright}(\Omega,\omega)$ be an ergodic p.m.p action.
For any $f\in L^{1}(\partial^{2}G\times\Omega,m\times\omega)$, for almost
every $(\xi,\eta,w)$ and any $a\in\mathbb{R}$:

$\lim_{b\to\infty}\frac{1}{b-a}\int_{\\{g\in
G|\tau(g,\xi,\eta)\in[a,b]\\}}f(g\xi,g\eta,gw)d\lambda(g)=\int
f(\xi,\eta,w)dm\times\omega$

We will then use this to prove theorem 1.6. Our methods will be closely based
on a paper by Bader and Furman [2], generalizing some of their results from
discrete to unimodular locally compact hyperbolic groups. While the proofs are
similar, the non-discreteness of the group makes the measure theoretic aspects
much more delicate.

###  1.4. Non unimodular groups

It is interesting to consider what happens in theorems 1.1 and 1.2 when the
unimodularity assumption is dropped. The following example shows that in
general theorem 7.2 dose not hold:

######  Example 1.8.

Consider the minimal parabolic subgroup $P$ of $SL_{2}(\mathbb{R})$, i.e the
group of $2$ by $2$ upper triangular matrices with determinant $1$. $P$ is
non-unimodular and non-elementary. This group is isomorphic to the group of
affine transformations of the real line with positive leading coefficient. The
Gromov boundary of this group is the boundary of the hyperbolic plane. In the
upper half plane model $P$ fixes $\infty$ and acts affinly on $\mathbb{R}$. We
shall see in 3.7 that any Patterson-Sullivan measure corresponding to any
metric on $P$ has no atoms and therefore is supported entirely on
$\mathbb{R}$. Since the action of $P$ on $\mathbb{R}$ is transitive there is a
unique $P$-invariant measure class on $\mathbb{R}$, the Lebesgue class. Thus
the corresponding representation is independent of the metric $d$ and
equivalent to the Koopman representation of $P$ on $L^{2}(\mathbb{R})$.
Because the leading coefficient of any affine map in $P$ is positive the
action on $L^{2}(\mathbb{R})$ preserves the subspaces of functions whose
Fourier transforms are supported on $\mathbb{R}_{+}$ and $\mathbb{R}_{-}$ so
this representation is not irreducible. For a full description of the
representation theory of $P$ see [14, Section 6.7].

Amenable hyperbolic locally compact groups are unimodular if and only if they
are elementary([10, Theorem 7.3]). Since $P$ is amenable it is very natural to
ask whether the unimodularity assumption in theorems 1.1 and 1.2 can be
replaced by the assumption that the group is non-amenable. We do not know the
answer to this question. Next we give an example of a non-amenable, non-
unimodular, hyperbolic locally compact group.

######  Example 1.9.

Let $T$ be a regular tree of degree $n<\infty$. Choose an orientation $O$ on
the edges of $T$ such that around each vertex, $k$ edges are oriented inward
and $n-k$ outward. consider the group of automorphisms of the tree preserving
the orientation, $G=Aut(T,O)$. If $k\neq n-k$ then $G$ is non-unimodular,
since the stabilizer of an edge has a different index in the stabilizers of
the corresponding vertices, even though the stabilizers of the two vertices
are conjugate. As long as $k,n-k\neq 1$, $G$ is non-amenable since it is non-
elementary and does not fix any end of the tree. As an example of the non-
unimodular non-amenable case it is interesting to ask whether $G$ satisfies
theorems 1.1 and 1.2. As far as we are aware it is not even known whether the
Koopman representation of $G$ corresponding to the standard measure on
$\partial T$ is irreducible.

###  1.5. Structure of the paper

In section 2 we fix the notation and our conventions for hyperbolic metric
spaces and we give an overview of the theory of locally compact hyperbolic
groups. In section 3 we develop the theory of Patterson-Sullivan measures for
locally compact hyperbolic groups. The theory is very similar to the discrete
case but has never been written down before. The main parts of the proof of
theorem 1.1 are given in section 4 and section 5. Section 4 Contains many of
the geometric aspects of the proof while section 5 includes the representation
theoretic sides. In section 6 we complete the proof of theorem 1.1 and in
section 7 we prove theorem 1.2 on rough equivalence of metrics. Section 8
contains applications to hyperbolic groups of type I. In section 9 and section
10 we construct a measurable version of a geodesic flow for $G$ and use it to
prove the ergodicity of the action on $\partial^{2}G$, a result which is
needed in section 7. In the appendices A and B we deal with measure theoretic
technicalities arising from the fact that our groups are non discrete.

###  1.6. Acknowledgments

I would like to thank my advisor, Uri Bader, for his constant support and
guidance. Every conversation with him leaves me in a better mood and in awe of
some amazing mathematics. Without him this project would never have come to
fruition. I would also like to thank my father, Yair Glasner, who has taught
me so much both in math and in general through the years and my mother,
Shalvia Glasner, for listening to me and helping me when things seemed hard
and unapproachable. Last but not least, I would like to thank my teammates at
the Weizmann institute: Alon Dogon, Itamar Vigdorovitch, Sheve Leibtag, Aviv
Taller, Benny Bachner, Paul Vollrath, Raz Slutsky, Gil Goffer, Tal Cohen, Guy
Salomon, Omer Lavi, Guy Kapon, Yuval Salant, Yuval Gorfine, Idan Pazi and
Peleg Bar-Sever for their constant friendship and support.

##  2\. Preliminaries

###  2.1. Notation and conventions

Throughout the paper we will use estimates involving additive and
multiplicative constants. In order to stop these constants from snowballing we
introduce the following notation. Given functions $f$ and $g$ with a common
domain, if there exists $0\leq c$ such that $f\leq g+c$ we write $f\lesssim
g$. If $f\lesssim g$ and $g\lesssim f$ we write $f\approx g$. Similarly, if
there exists $0<C$ such that $f\leq Cg$ we write $f\prec g$ and if $f\prec g$
and $g\prec f$ we write $f\asymp g$. If the constants in an estimate depend on
a parameter we write the parameter as subscript in the inequality to denote
the fact that the estimate is not uniform in that parameter. For example
$f(x,y)\prec_{y}g(x,y)$ means that there exists $0<C=C(y)$, possibly depending
on $y$ but not depending on $x$ such that $f(x,y)\leq C(y)g(x,y)$. The same
conventions are used in [15].

###  2.2. Maps between metric spaces

Given two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ we call a function
$f:X\rightarrow Y$ an (L,c)-quasi-isometric embedding if

$\frac{1}{L}d_{X}(x_{1},x_{2})-c\leq d_{Y}(f(x_{1}),f(x_{2})\leq
Ld_{X}(x_{1},x_{2})+c$

A (L,c)-quasi-isometric embedding is called a (L,c)-quasi-isometry if there
exists $0\leq D$ such that for any $y\in Y$ there exists $x\in X$ such that
$d_{Y}(f(x),y)\leq D$. A (1,c)-quasi-isometric embedding is called a c-rough
embedding and a (1,c)-quasi-isometry is called a c-rough isometry. A map
$f:X\rightarrow Y$ is a (L,c)-rough similarity if $Ld_{X}(x_{1},x_{2})-c\leq
d_{Y}(f(x_{1},x_{2})\leq Ld_{X}(x_{1},x_{2})+c$. When the constants are
irrelevant we will omit them and say ”f is a quasi-isometry” or ”f is a rough
embedding”.

A geodesic in a metric space X is an isometric embedding of $\mathbb{R}$. We
define quasi-geodesics and rough geodesics as quasi-isometric embeddings and
rough embeddings of $\mathbb{R}$. We make the same definitions for rays and
segments replacing $\mathbb{R}$ by closed rays and closed intervals in
$\mathbb{R}$. A metric space is called geodesic if every two points can be
connected by a geodesic segment. Similarly a metric space is called
(L,c)-quasi-geodesic if any two points can be connected by a (L,c)-quasi-
geodesic segment and c-roughly geodesic if it is (1,c)-quasi-geodesic. Here
also we will omit the constants if they are irrelevant. Given a quasi-geodesic
or roughly geodesic space we will implicitly assume that all quasi-geodesics
or rough geodesics used have uniformly bounded constants compatible with the
constants of the space.

###  2.3. Hyperbolic metric spaces

A good introduction to the theory of hyperbolic metric spaces can be found in
[7, Chapter 3.H]. A more general theory for non-proper non-geodesic spaces is
developed in [13]. We provide here a brief survey.

Let (X,d) be a metric space. Given $x,y,z\in X$ define the Gromov product of
$x$ and $y$ with respect to $z$:

$(x,y)_{z}=\frac{1}{2}(d(z,x)+d(z,y)-d(x,y))$

Note that $|(x,y)_{z}-(x,y)_{w}|\leq d(w,z)$. If $X$ is equipped with a base
point $o$ we will denote $(x,y)=(x,y)_{o}$

######  Definition 2.1.

Let $0\leq\delta$. A metric space (X,d) is called $\delta$-hyperbolic if for
any $x,y,z,o\in X$

$(x,z)_{o}\geq\min\\{(x,y)_{o},(y,z)_{o}\\}-\delta$

A metric space is (Gromov)-hyperbolic if it is $\delta$-hyperbolic for some
$0\leq\delta$.

Given a hyperbolic space $X$ fix a base point $o$. A sequence $(x_{n})$ is
called Cauchy-Gromov if $\lim_{n,m\to\infty}(x_{n},x_{m})=\infty$. We say such
a sequence converges to infinity. Two such sequences $(x_{n}),(y_{n})$ are
said to be equivalent if $\lim_{n\to\infty}(x_{n},y_{n})=\infty$. Using
hyperbolicity of the space one sees that this is an equivalence relation.
These notions don’t depend on $o$.

######  Definition 2.2.

The Gromov boundary of a hyperbolic space $X$ is

$\partial X=\\{[(x_{n})]|(x_{n})\text{is Cauchy-Gromov}\\}$

We denote $\bar{X}=X\cup\partial X$.

We now extend the definition of the Gromov product to points in $\bar{X}$.
Given $\xi,\eta\in\partial X$ and $z,o\in X$ define:

$(\xi,\eta)_{o}=\inf\liminf_{n\to\infty}(x_{n},y_{n})_{o}$
$(\xi,z)_{o}=\inf\liminf_{n\to\infty}(x_{n},z)_{o}$

where the infimum is taken over sequences representing the boundary points.
After replacing $\delta$ by $2\delta$ the extension of the Gromov product
still satisfies the condition given in the definition of hyperbolicity. It
follows from hyperbolicity that

$|\sup\limsup_{n\to\infty}(x_{n},y_{n})_{o}-\inf\liminf_{n\to\infty}(x_{n},y_{n})_{o}|<2\delta$

$|\sup\limsup_{n\to\infty}(x_{n},z)_{o}-\inf\liminf_{n\to\infty}(x_{n},z)_{o}|<2\delta$

Thus up to $2\delta$ one can compute the Gromov product by any choice of
sequences representing the points. We will only care about the Gromov product
up to an additive constant so this will be useful.

$\bar{X}$ can be given a topology as follows. Choose a base point $o$ for $X$.
Open sets in $X$ are just the open sets in the metric topology on $X$. A
family of basic (not necessarily open) neighborhoods around a point
$\xi\in\partial X$ are given by $\\{x\in\bar{X}|(x,\xi)>M\\}$ where
$M\in[0,\infty)$. In this topology $x_{n}$ converges to $\xi\in\partial X$ if
and only if $(x_{n},\xi)\rightarrow\infty$. This topology does not depend on
the base point $o$ and if $X$ is a proper geodesic metric space then $\partial
X$ and $\bar{X}$ are compact.

######  Lemma 2.3.

The Gromov product is lower semi continuous as a function
$(\cdot,\cdot)_{\cdot}:\bar{X}\times\bar{X}\times
X\rightarrow\mathbb{R}\cup\\{\infty\\}$.

The proof can be found in [13, Lemma 3.4.23]

######  Definition 2.4.

Let $X$ be a hyperbolic metric space. A visual metric $d_{\varepsilon}$ with
parameter $\varepsilon>0$ on $\partial X$ is a metric satisfying:

$d_{\varepsilon}(\xi,\eta)\asymp e^{-\varepsilon(\xi,\eta)}$ (1)

For small enough $\varepsilon$ a visual metric always exists (see [13,
Proposition 3.6.8], [7, Chapter 3.H.3]). Such a metric always generates the
standard topology on $\partial X$. The hyperbolicity implies that a visual
metric $d_{\varepsilon}$ is almost an ultra-metric in the following sense, for
all $\xi_{1}$, $\xi_{2}$, $\xi_{3}$:

$d_{\varepsilon}(\xi_{1},\xi_{3})\prec
max\\{d_{\varepsilon}(\xi_{1},\xi_{2}),d_{\varepsilon}(\xi_{2},\xi_{3})\\}$

######  Definition 2.5.

Let X be a hyperbolic metric space, $x,o\in X$ and $\sigma>0$. The shadow cast
by $x$ from $o$ with parameter $\sigma$ on $\partial X$ is

$\Sigma_{o}(x,\sigma)=\\{\xi\in\partial X|(x,\xi)_{o}+\sigma>d(x,o)\\}$

If o is a given base point of X we will omit it from the notation.

We now mention three lemmas we will need in order to work with hyperbolic
locally compact groups. These lemmas are standard for geodesic hyperbolic
spaces but we will need them in a more general setting. The proofs of these
lemmas are based on the ones given in [15, Subsection 3.1] and are basically a
restating of results from [5] and [6]

######  Lemma 2.6.

Let $X$ be a geodesic hyperbolic metric space and $Y$ a metric space which is
quasi-isometric to $X$, then $Y$ is hyperbolic if and only if it is roughly
geodesic.

###### Proof.

We follow the proof in [15, Subsection 3.1]. By [5, Theorem A.1] $Y$ is
hyperbolic if and only if $Y$ is quasi-ruled in the sense defined in [5]. By
[5, Lemma A.2] a quasi-ruled space is roughly geodesic, on the other hand a
roughly geodesic space is obviously quasi-ruled. ∎

######  Lemma 2.7.

Let X be a roughly geodesic hyperbolic space then:

  1. (1)

For any $x\in X$ and $\xi\in\partial X$ there is a roughly geodesic ray
$\gamma$ with $\gamma(0)=x$, $\gamma(\infty)=\lim_{t\to\infty}\gamma(t)=\xi$.

  2. (2)

For any $\xi,\eta\in\partial X$ there is a rough geodesic $\gamma$ with
$\gamma(-\infty)=\xi$, $\gamma(\infty)=\eta$.

###### Proof.

This follows from [6, Proposition 5.2] and from the obvious fact that roughly
geodesic spaces are almost geodesic in the sense defined in [6]. Note that as
a consequence of [6, Proposition 5.2] the almost geodesic hyperbolic spaces
are precisely the roughly geodesic ones. ∎

######  Lemma 2.8.

If $f:X\to Y$ is a quasi-isometry of roughly geodesic hyperbolic spaces, $o\in
X$, then there exists $L,C>0$ such that
$\frac{1}{L}(x,y)-C\leq(f(x),f(y))_{f(o)}\leq L(x,y)_{o}+C$. As a result $f$
canonically extends to a homeomorphism $\partial f:\partial X\to\partial Y$.

###### Proof.

The estimate on Gromov products is exactly [6, Proposition 5.5]. It follows
that $f$ respects the notion of being Cauchy-Gromov so $f$ extends to the
boundary, furthermore the estimate implies this extension is continuous.
Applying the same argument to a quasi-inverse of $f$ shows the map between
boundaries is a homeomorphism. ∎

Finally we will need one last lemma.

######  Lemma 2.9.

Let $X$ be a roughly geodesic hyperbolic space with base point $o$, fix a
visual metric $d_{\epsilon}$ on $\partial X$. Let $C\geq 0$. For large enough
$\sigma$ (depending on $C$), for any $\xi\in\partial X$, $R>0$, any roughly
geodesic ray $\gamma$ with $\gamma(0)=o,\gamma(\infty)=\xi$ and for any $g$
such that $d(\gamma(R),g)\leq C$:

$B_{e^{-\epsilon R}}(\xi)\subseteq\Sigma_{o}(g,\sigma)$

In particular for large enough $\sigma$:

$B_{e^{-\epsilon R}}(\xi)\subseteq\Sigma_{o}(\gamma(R),\sigma)$

###### Proof.

On the one hand $(\gamma(R),\xi)\approx|\gamma(R)|\approx R$. On the other
hand if $d_{\epsilon}(\xi,\eta)<e^{-\epsilon R}$ then $(\xi,\eta)\gtrsim R$
and since $d(\gamma(R),g)<C$ , $(\gamma(R),g)\approx_{C}R$. Now we get
$(g,\eta)\gtrsim
min\\{(g,\gamma(R)),(\gamma(R),\xi),(\xi,\eta)\\}\approx_{C}R$. Thus if
$\sigma$ is chosen large enough (independently of $o,\xi,R,\gamma$ but
depending on $C$) we get $\eta\in\Sigma_{o}(\gamma(R),\xi)$. Therefore
$B_{e^{-\epsilon R}}(\xi)\subseteq\Sigma_{o}(g,\sigma)$ as needed. ∎

###  2.4. Locally compact hyperbolic groups

This subsection will present the framework of locally compact hyperbolic
groups in which we will work. A good introduction to the concept of a locally
compact hyperbolic group can be found in [10]. We will use the theory
developed there.

######  Definition 2.10.

A locally compact group $G$ is called hyperbolic if it is compactly generated
and for some (hence any) compact generating set the induced word metric on $G$
is hyperbolic.

In the discrete case this definition recovers the standard notion of
hyperbolicity for groups. By [10, Corollary 2.6] $G$ is hyperbolic if and only
if there exists a proper geodesic metric space $X$ on which $G$ admits a
continuous proper co-compact isometric action. The action of G extends
continuously to $\bar{X}$. We obtain the following examples:

  1. (1)

Rank one simple Lie groups with finite center are hyperbolic.

  2. (2)

Minimal parabolic subgroups of rank one simple Lie groups with finite center
are hyperbolic.

  3. (3)

Groups acting properly and co-compactly on locally finite trees are
hyperbolic.

  4. (4)

Generalizing the previous examples, locally compact groups acting
isometrically properly and co-compactly on proper CAT(-1) spaces are
hyperbolic.

  5. (5)

Totally disconnected compactly generated groups are hyperbolic if and only if
some (and hence every) associated Cayley-Abels graph is hyperbolic.

Let $G$ be a locally compact hyperbolic group. Denote by $\mathcal{D}(G)$ the
set of all left invariant hyperbolic metrics on $G$ which are quasi-isometric
to a word metric on G and are Borel measurable as functions $d:G\times
G\rightarrow\mathbb{R}$. If $d\in\mathcal{D}(G)$ we will always consider
$(G,d)$ to be equipped with base point $1\in G$. Since word metrics are
roughly isometric to the corresponding Cayley graph which is a geodesic space,
it follows from 2.6 that all the metrics in $\mathcal{D}(G)$ are roughly
geodesic. In addition by 2.8 we see that all such metrics give rise to the
same boundary which we shall denote $\partial G$. Similarly if $G$ acts on $X$
as above, $\partial G\cong\partial X$. The following are examples of metrics
in $\mathcal{D}(G)$:

  1. (1)

Word metrics corresponding to compact generating sets.

  2. (2)

If $G$ acts on $(X,d_{X})$ as above, for any $x\in X$ one can define
$d_{0}(g,h)=d_{X}(gx,hx)$. In general this is only a pseudo-metric but any
left invariant Borel measurable metric roughly similar to it will be in
$\mathcal{D}(G)$. There always exists such a metric, for example one can
define $d(g,g)=0,d(g,h)=d_{0}(g,h)+1$ for $g\neq h$. Alternatively one can
adjust the theory to work with pseudo-metrics.

Note that we do not require the metrics in $\mathcal{D}(G)$ to be continuous,
the reason for this is that we will only be interested in the coarse geometry
of the group and won’t care about local properties.

######  Lemma 2.11.

Let $G$ be a locally compact hyperbolic group, then $\partial G$ is compact
and the action of $G$ is continuous.

###### Proof.

By [10, Proposition 2.1] there exists a space $X$ with a $G$ action as above.
Since $X$ is proper and geodesic $\partial X$ is compact and the action on
$\partial X$ is continuous, but $\partial G\cong\partial X$. ∎

The following lemma is very important and expresses algebraically the
contracting dynamics of $G$ on $\partial G$:

######  Lemma 2.12.

Let $G$ be a locally compact hyperbolic group and $d\in\mathcal{D}(G)$ then
for $g\in G$ and $\xi\in\overline{G}$:

$(g,g\xi)\approx|g|-(g^{-1},\xi)$

uniformly in $g$ and $\xi$.

###### Proof.

The proof is taken from [15, Lemma 4.1]. If $h\in G$ then a direct calculation
shows that $(g,gh)=|g|-(g^{-1},h)$. Letting $h$ converge to $\xi\in\bar{G}$ we
get the desired result. ∎

We get the following corollary:

######  Lemma 2.13.

([15, Lemma 4.1]) Let $G$ be a locally compact hyperbolic group and
$d\in\mathcal{D}(G)$ then for $g\in G$:

$|g|\approx\sup_{\xi\in\partial G}(g,\xi)$

uniformly in $g$.

###### Proof.

The proof is taken from [15, Lemma 4.1]. Obviously $(g,\xi)\leq|g|$ for all
$g$. Let $\xi_{1},\xi_{2}\in\partial G$ be two distinct points. By 2.12 for
any $g\in G$:

$\max_{i}(g,g\xi_{i})\approx|g|-\min_{i}(g^{-1},\xi_{i})$

But $(\xi_{1},\xi_{2})\gtrsim\min\\{(g^{-1},\xi_{1}),(g^{-1},\xi_{2})\\}$, so
for some $i$, $(g,g\xi_{i})\approx|g|$.

∎

Let $d\in\mathcal{D}(G)$. Even though the metric topology induced on $G$ by
$d$ need not be proper or geodesic the space $\bar{G}=G\cup\partial G$ can be
given a compact topology agreeing with the locally compact topology on $G$.
Furthermore this topology does not depend on $d$. To do this one sets a base
of (not necessarily open) neighborhoods of $\xi\in\partial G$ to be
$\\{x\in\bar{G}|(x,\xi)>M\\}$. So the topology is the regular topology on $G$
and $x_{n}\to\xi$ if and only if $(x_{n},\xi)\to\infty$.

######  Lemma 2.14.

The topology described above is independent of $d$ and compact.

###### Proof.

Let $\\{U_{i}\\}_{i\in I}$ be an open cover of $\bar{G}$. The topology on
$\partial G$ is the same as the standard one because the topology described is
easily seen to be the one induced by the visual metric. Since $\partial G$ is
compact there exists a finite set $\\{U_{i}\\}_{i\in J}$ covering $\partial
G$. For each $i\in J$ there exists $\xi_{i}\in\partial G$ and $M_{i}>0$ such
that $\\{x\in\bar{G}|(x,\xi_{i})>M_{i}\\}\subseteq U_{i}$. If now $g\notin
U_{i}$ and $\eta\in U_{i}$ then $M_{i}\gtrsim(g,\eta)$ by hyperbolicity.
Therefore if $g\notin U_{i}$ for all $i\in J$ then there exists $M>0$ such
that $(g,\eta)<M$ for all $\eta\in\partial G$. Now by 2.13
$|g|\approx\sup_{\xi\in\partial G}(g,\xi)$ so $G\backslash\bigcup_{i\in
J}U_{i}$ is bounded in the metric $d$ and therefore has compact closure in
$G$. Thus the closure of $G\backslash\bigcup_{i\in J}U_{i}$ can be covered by
finitely many of the remaining $U_{i}$ as needed.

To see that the topology is independent of $d$ notice that if $\xi\in\partial
G$ then $g_{n}\to\xi$ if and only if $(g_{n},\xi)\to\infty$ and by 2.8 this
condition is independent of $d$. ∎

By [13, Theorem 6.1.4],[10, Section 3] elements in $G$ can be classified into
three types:

  1. (1)

Elliptic elements: elements with bounded orbits.

  2. (2)

Parabolic elements: non-elliptic elements with one fixed point in $\partial
G$.

  3. (3)

Hyperbolic elements: non-elliptic elements with two fixed points in $\partial
G$. One of the fixed points is attracting and the other is repelling. If
$\xi\in\bar{G}$ is not the repelling fixed point of a hyperbolic element $g$
then $g^{n}\xi$ converges to the attracting fixed point.

In the discrete case there are no parabolic elements but in the locally
compact case they can appear.

By [13, Section 3] [1, Theorem 6.8] locally compact hyperbolic groups can be
classified in to three types:

  1. (1)

Elementary: groups with finite boundary. In this case either $\partial G=\phi$
and $G$ is compact or $|\partial G|=2$ and $G$ contains an infinite cyclic co-
compact subgroup.

  2. (2)

Non-elementary amenable: non-elementary groups stabilizing a point in
$\partial G$. These groups have uncountable boundary and are amenable by [1,
Theorem 6.8].

  3. (3)

General type: groups not stabilizing any point in $\partial G$. These groups
have uncountable boundary and are never amenable.

Being of general type is equivalent to being non-amenable. By [10, Theorem
7.3] if a non-elementary hyperbolic locally compact group is amenable then it
is never unimodular. Thus unimodular locally compact hyperbolic groups are
either elementary or of general type.

Non-elementary amenable groups do not exist in the discrete case but in the
locally compact case they can appear. Examples of such groups are minimal
parabolic subgroups of rank-$1$ simple Lie groups with finite center.

######  Theorem 2.15.

If $G$ is hyperbolic locally compact of general type then the action on
$\partial G$ is minimal.

###### Proof.

By [13, Theorem 7.4.1] any closed invariant subset of $\partial G$ containing
at least two points is all of $\partial G$ but since $G$ is of general type
there is no fixed point in $\partial G$. ∎

Finally we will need the following lemma:

######  Lemma 2.16.

Let $G$ be hyperbolic locally compact of general type. Any $G$-equivariant
continuous map $f:\partial G\to\partial G$ is trivial.

###### Proof.

Since $G$ is of general type, $G$ contains hyperbolic elements ([10, Section
3]). The set of all fixed points in $\partial G$ of hyperbolic elements is
invariant and non-empty so since the action is minimal it is dense. If
$f:\partial G\to\partial G$ is $G$-equivariant and continuous then $f$ must
stabilize the attracting fixed point and the repelling fixed point of any
hyperbolic element, since continuity and equivariance imply that if $\xi$ is
attracting or repelling for $g$ so is $f(\xi)$. Thus $f$ fixes a dense set. ∎

We add one last remark on hyperbolic locally compact groups. It is not clear
to us that given $d\in\mathcal{D}(G)$ the corresponding Gromov product
$(\cdot,\cdot):\bar{G}\times\bar{G}\to\mathbb{R}$ is measurable, however one
can always slightly modify it to be measurable. We will only be interested in
the values of the Gromov product up to a uniform constant since we are only
interested in the coarse geometric structure of $d$. Therefore we will change
the definition of the Gromov product in a way that changes the values only by
a bounded amount and ensures that it is measurable. Choose a countable subset
$S\subseteq G$ which is co-bounded, i.e any element is a uniformly bounded
distance away from an element of $S$. The space $(S,d)$ is hyperbolic and
roughly isometric to $(G,d)$ so they give rise to the same boundary. Since the
Gromov product is lower semi-continuous on $\bar{S}$ it is Borel measurable
(this is also true on $G$ but the Borel $\sigma$-algebra will be the one
corresponding to the metric $d$ and not to the standard topology on $G$, this
is the reason for this entire remark). Denote the Gromov product on $\bar{S}$
by $\langle\cdot,\cdot\rangle$. Choose a measurable closest point projection
$P:G\to S$ and extend it to the boundary via the identity. Now if
$x,y,z\in\bar{G}$ the function $\langle P(x),P(y)\rangle_{P(z)}$ is Borel
measurable on $G$ and at bounded distance from the original definition of the
Gromov product on $G$, so we can take it to be the new Gromov product. We will
make one more change for convenience: we assume that the formula
$(g,h)_{o}=\frac{1}{2}(|g|+|h|-d(g,h))$ still holds when $g,h,o\in G$.

###  2.5. Non-singular actions and Koopman representations

Consider a locally compact group $G$ with a measurable non-singular (i.e.
measure class preserving) action on a measure space $(X,\mu)$. One can define
an associated unitary representation of $G$ called the Koopman representation
on $L^{2}(X,\mu)$ by:

$[\pi(g)f](\xi)=\sqrt{\frac{dg_{*}\mu}{d\mu}(\xi)}f(g^{-1}\xi)$

The Koopman representation depends only on the measure class $[\mu]$ and not
on the actual measure. If $\nu$ is in the same measure class as $\mu$ then the
map $L^{2}(X,\nu)\to L^{2}(X,\mu)$ defined by
$f\mapsto\sqrt{\frac{d\mu}{d\nu}}f$ is an isomorphism of the Koopman
representations corresponding to $\mu,\nu$.

By [4, Proposition A.6.1], if $G$ is $\sigma$-compact and $(X,\mu)$ is a
standard measure space this representation is continuous with respect to the
strong operator topology. If $G$ is locally compact hyperbolic then $G$ is
compactly generated hence $\sigma$-compact so the above theorem holds.

##  3\. Patterson-Sulivan Measures

In this section we will construct from a metric $d\in\mathcal{D}(G)$ an
invariant measure class $[\mu]$ on $\partial G$ called the Patterson-Sullivan
measure class associated to $d$. The original construction of such measures in
due to Patterson in the case of Fuchsian groups [17]. Sullivan later extended
this to the case of discrete subgroups of $SO(n,1)$ [19]. Further
generalizations were made by Coornaert for discrete hyperbolic groups [12],
and by Burger and Mozes for non-discrete groups acting on CAT(-1) spaces [8].
We give a construction for general hyperbolic locally compact groups.

Fix a locally compact hyperbolic group $G$, $d\in\mathcal{D}(G)$, a left Haar
measure $\lambda$ on $G$ and a visual metric $d_{\varepsilon}$.

######  Definition 3.1.

The critical exponent of (G,d) is

$h=\inf\\{s|\int\\!e^{-s|g|}d\lambda(g)<\infty\\}=\limsup_{r\to\infty}\frac{1}{r}\lambda(B_{r}(1))$

$h$ is the exponential growth rate of $G$ with respect to $d$. If $G$ is non-
elementary then by [10, Section 3] $G$ contains a discrete free quasi
isometrically embedded (Schottky) subgroup or subsemigroup and thus $h>0$. On
the other hand, if $G$ is elementary then obviously $h=0$. We denote
$D=\frac{h}{\epsilon}$.

######  Definition 3.2.

A finite measure $\mu$ on $\partial G$ is called quasi-conformal of dimension
$\frac{\alpha}{\epsilon}$ if for all $g\in G$, for $\mu$ almost every $\xi$

$\frac{dg_{*}\mu}{d\mu}(\xi)\asymp e^{-\alpha(|g|-2(g,\xi))}$

The quantity $|g|-2(g,\xi)$ in the above definition is the analogue of the
Busemann function (see [7, Definition II.8.17]) corresponding to $\xi$, i.e.
it roughly measures the ”distance from $\xi$” normalized to be $0$ at $1$. We
will not use this formally but it might help understand the geometric picture.

######  Theorem 3.3.

If G is non-compact, there exists a quasi-conformal measure of dimension
$D=\frac{h}{\epsilon}$ on $\partial G$.

We will later see that If G is non-elementary, the measure class of such a
measure is uniquely determined. This measure class $[\mu]$ is also known as
the Patterson-Sullivan measure class. After showing the measure class is
unique we will fix such a measure $\mu$ and call it the Patterson-Sullivan
measure of $(G,d)$.

###### Proof.

There are many similar proofs in the literature. We imitate the proof in [12,
Theorem 5.4], another example is [8, Section 1]. Recall that the space
$\bar{G}$ has a natural compact topology on it described in 2.14. By [8, Lemma
1.2] There exists an increasing continuous function
$H:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ satisfying

  1. (1)

$\int_{G}H(e^{|x|})e^{-s|x|}d\lambda(x)$ converges for $s>h$ and diverges at
$s=h$.

  2. (2)

$\forall\alpha>0,\exists t_{0}>0$ such that $\forall t>t_{0}$ and $k>1$,
$H(kt)\leq k^{\alpha}H(t)$.

If one assumes that $\int e^{-h|x|}d\lambda(x)$ diverges one can take $H=1$
and the proof becomes simpler. Denote

$\mathcal{P}(s)=\int_{G}H(e^{|x|})e^{-s|x|}d\lambda(x)$

and define probability measures on $\bar{G}$ by:

$d\mu_{s}=\frac{1}{\mathcal{P}(s)}H(e^{|x|})e^{-s|x|}d\lambda(x)$

Let $\mu$ be a weak-$*$ limit point of $\mu_{s}$ as $s\searrow h$. We claim
$\mu$ is supported on $\partial G$ and is quasi-conformal of dimension $D$.

To see that $\mu(G)=0$ notice that since $\mathcal{P}(s)\to\infty$ when
$s\searrow h$, $\mu_{s}(K)\to 0$ for every compact $K\subset G$.

In order to show that $\mu$ is quasi-conformal of dimension $D$ it is enough
to show that for every $g\in G$ and $\xi\in\partial G$ there exists a
neighborhood $V$ of $\xi$ in $\bar{G}$ such that for any continuous function
$f$ supported in $V$:

$\int f(x)dg_{*}\mu(x)\asymp e^{-h(|g|-2(g,\xi))}\int f(x)d\mu(x)$

uniformly in $g,\xi$. Indeed let $g\in G$, we have:

$\frac{dg_{*}\mu_{s}}{d\mu_{s}}(x)\asymp\frac{H(e^{|g^{-1}x|})}{H(e^{|x|})}e^{-s(|g^{-1}x|-|x|)}=\frac{H(e^{|g^{-1}x|})}{H(e^{|x|})}e^{-s(|g|-2(g,x))}$

Let $\xi\in\partial G$. If $x\in G$ such that $(x,\xi)>|g|$ then by
hyperbolicity $(g,x)\approx(g,\xi)$ uniformly in $g,\xi$. In addition since
$||x|-|g^{-1}x||<|g|$ it follows from the second property of $H$ that there
exists $C=C(|g|)>0$ such that if $|x|>C$ then
$\frac{H(e^{|g^{-1}x|})}{H(e^{|x|})}\asymp 1$. Let now $M=max\\{C,|g|\\}$ and
consider $V=\\{x\in\bar{G}|(x,\xi)>M\\}$. By definition $V$ contains an open
neighborhood of $\xi$. If $x\in V$ then in particular $|x|>M$. Therefore by
the formula for $\frac{dg_{*}\mu_{s}}{d\mu_{s}}$ and since
$(g,x)\approx(g,\xi)$ we get that for $x\in V$,
$\frac{dg_{*}\mu_{s}}{d\mu_{s}}(x)\asymp e^{-h(|g|-2(g,\xi))}$. Integrating
any continuous function $f$ supported on $V$ we get:

$\int f(x)dg_{*}\mu_{s}=\int
f(x)\frac{dg_{*}\mu_{s}}{d\mu_{s}}(x)d\mu_{s}\asymp\int
f(x)e^{-h(|g|-2(g,\xi))}d\mu_{s}$

Taking a partial limit $s\searrow h$ we get that the same holds for $\mu$ and
thus $\mu$ is quasi-conformal of dimension $D$ as needed.

∎

######  Lemma 3.4 (Sullivan’s shadow lemma).

If $\mu$ is quasi-conformal of dimension $D>0$, is not supported on a single
atom and $\sigma$ is large enough then

$\mu(\Sigma(g,\sigma))\asymp_{\sigma}e^{-h|g|}$

Note that if $\mu$ is supported on a single atom then it must be invariant and
this can only happen if $h=0$ i.e. if $G$ is elementary.

###### Proof.

Let $g\in G$, we have

$\displaystyle\mu(g^{-1}\Sigma(g,\sigma))=$
$\displaystyle\int_{\Sigma(g,\sigma)}\frac{dg_{*}\mu}{d\mu}(\xi)d\mu(\xi)$
$\displaystyle\asymp$
$\displaystyle\int_{\Sigma(g,\sigma)}e^{-h(|g|-2(g,\xi))}d\mu(\xi)$
$\displaystyle\asymp$ ${}_{\sigma}\int_{\Sigma(g,\sigma)}e^{h|g|}d\mu(\xi)$
$\displaystyle=$ $\displaystyle\mu(\Sigma(g,\sigma))e^{h|g|}$

Now in order to finish the proof we must bound $\mu(g^{-1}\Sigma(g,\sigma))$
independently of $g$. Let $t<1$ be larger than the size of the largest atom of
$\mu$. There exists $\delta>0$ such that the measure of any ball of radius
less then $\delta$ is less than $t$. Indeed, if $B_{r_{n}}(\xi_{n})$ is a
sequence of balls with $r_{n}\to 0$ and $\mu(B_{r_{n}}(\xi_{n}))\geq t$ after
passing to a convergent sub-sequence we can assume $\xi_{n}\to\xi$. For any
$r>0$, $B_{r}(\xi)$ contains a ball of the form $B_{r_{n}}(\xi_{n})$ so
$\mu(B_{r}(\xi))\geq t$ and thus $\mu(\\{\xi\\})\geq t$ which is a
contradiction.

Now the diameter of $\partial G\backslash g^{-1}\Sigma(g,\sigma)$ converges to
$0$ as $\sigma\to\infty$ uniformly in g. To see this, notice that if
$\xi\notin g^{-1}\Sigma(g,\sigma)$ then $g\xi\notin\Sigma(g,\sigma)$ so
$(g,g\xi)\leq|g|-\sigma$. By lemma 2.12 we see that
$(\xi,g^{-1})\gtrsim\sigma$. So if $\xi,\eta\notin g^{-1}\Sigma(g,\sigma)$
then:

$(\xi,\eta)\gtrsim min\\{(\xi,g^{-1})(\eta,g^{-1})\\}\gtrsim\sigma$

Therefore $d_{\varepsilon}(\xi,\eta)\prec e^{-\sigma}$ as needed. So for
$\sigma$ large enough we have that $1\geq\mu(g^{-1}\Sigma(g,\sigma))>1-t$ and
thus
$\mu(\Sigma(g,\sigma))\asymp_{\sigma}e^{-h|g|}\mu(g^{-1}\Sigma(g,\sigma))\asymp
e^{-h|g|}$ finishing the proof. ∎

We assume from now on that $G$ is non-elementary so $h>0$ and the shadow lemma
holds. Denote $A_{R}(\alpha)=\\{g\in G|R-\alpha<|g|<R+\alpha\\}$, the annulus
of radius $R$ and thickness $\alpha$ around the identity in $G$. For our
purposes the parameter $\alpha$ should be thought of as an arbitrary parameter
which we will take to be as large as we need and then keep it fixed.

######  Lemma 3.5.

For $\alpha$, $\sigma$ large enough and any $\xi\in\partial G$,
$\lambda(\\{g\in
A_{R}(\alpha)|\xi\in\Sigma(g,\sigma)\\})\asymp_{\sigma,\alpha}1$. i.e , The
shadows of elements of the annuli $A_{R}(\alpha)$ cover $\partial G$ with
”bounded multiplicity”.

###### Proof.

Let $\xi\in\partial G$ and $\gamma$ a roughly geodesic ray with
$\gamma(0)=1,\gamma(\infty)=\xi$. Fix $r>0$ large enough so that the measure
of an $r$-ball in G is positive, and choose $\alpha$ large enough so that
$B_{r}(\gamma(R))\subseteq A_{R}(\alpha)$. For any $g\in B_{r}(\gamma(R))$,
$(g,\xi)\approx_{r}|g|$ and thus for $\sigma$ large enough, and independent of
$g,\xi,R$ we have $\xi\in\Sigma(g,\sigma)$. Thus $\lambda(\\{g\in
A_{R}(\alpha)|\xi\in\Sigma(g,\sigma)\\}\geq\lambda(B_{r}(\gamma(R)))$.

On the other hand if $g,h\in A_{R}(\alpha)$ and
$\xi\in\Sigma(g,\sigma)\cap\Sigma(h,\sigma)$ then since
$|g|\approx_{\alpha}|h|\approx_{\alpha}R$:

$(g,h)\gtrsim_{\alpha,\sigma}min\\{(g,\xi),(\xi,h)\\}\approx_{\alpha,\sigma}R$

Using again that $|g|\approx_{\alpha}|h|\approx_{\alpha}R$ and plugging this
in to the definition of the Gromov product we deduce that
$d(g,h)\approx_{\alpha,\sigma}0$. Therefore $\\{g\in
A_{R}(\alpha)|\xi\in\Sigma(g,\sigma)\\}$ is contained in a ball with radius in
independent of $g,R,\xi$ giving the other inequality. ∎

We will also need the following discrete version of lemma 3.5:

######  Lemma 3.6.

Fix $C>0$. For large enough $\alpha$, large enough $\sigma$ (perhaps depending
on $C$) and any $R>0$, there exists a finite subset $F\subseteq A_{R}(\alpha)$
such that the shadows $\\{\Sigma(g,\sigma)|g\in F\\}$ cover $\partial G$ and
for any $g,h\in F$, $d(g,h)>C$.

###### Proof.

Choose $\alpha$ large enough that for any roughly geodesic ray $\gamma$ with
$\gamma(0)=1$, $\gamma(R)\in A_{R}(\alpha)$. Since $\partial G$ is compact
there exists a finite set of points $\xi_{i}\in\partial G$ such that the balls
of radius $e^{-\epsilon R}$ cover $\partial G$. Choose roughly geodesic rays
$\gamma_{i}$ such that $\gamma_{i}(0)=1$ and $\gamma_{i}(\infty)=\xi_{i}$. By
lemma 2.9, for some $\sigma>0$ (depending on $C$), if
$d(\gamma_{i}(R),\gamma_{j}(R))\leq C$ then $B_{e^{-\epsilon
R}}(\xi_{j})\subseteq\Sigma(\gamma_{i}(R),\sigma)$. Let $F$ be a maximal
subset of the $\gamma_{i}(R)$ with the property that for all $g,h\in F$,
$d(g,h)>C$ then $\\{\Sigma(g,\sigma)|g\in F\\}$ covers $\partial G$ and has
the desired properties.

∎

The only use we will make of the parameter $\sigma$ is in lemma 2.9, lemma
3.4, lemma 3.5 and in lemma 3.6, furthermore we will only use lemma 2.9 and
lemma 3.6 for a specific constant $C$ in the proof of lemma 5.6 depending only
on $(G,d)$. All three lemmas only hold for $\sigma$ which is large enough. We
fix such a $\sigma$ and denote $\Sigma(g)=\Sigma(g,\sigma)$ for every $g\in
G$. After this we will have no more need to mention the parameter $\sigma$
explicitly again. We state what we will use from lemma 2.9, lemma 3.4, lemma
3.5 and lemma 3.6:

  * •

For every $g\in G$, $R>0$ and roughly geodesic ray $\gamma$ with
$\gamma(0)=1,\gamma(\infty)=\xi$: $B_{e^{-\epsilon
R}}(\xi)\subseteq\Sigma(g)$.

  * •

For every $g\in G$: $\mu(\Sigma(g))\asymp e^{-h|g|}$.

  * •

For large enough $\alpha$ and any $\xi\in\partial G$, $\lambda(\\{g\in
A_{R}(\alpha)|\xi\in\Sigma(g)\\})\asymp_{\alpha}1$.

  * •

For a fixed constant $C$ from the proof of 5.6 which depends only on $(G,d)$,
for large enough $\alpha$ and for any $R>0$ there exists a finite $F\subseteq
A_{R}(\alpha)$ such that the shadows $\\{\Sigma(g)|g\in F\\}$ cover $\partial
G$ and for any $g,h\in F$, $d(g,h)>C$.

An immediate consequence of the shadow lemma (or alternatively lemma 2.12) is
that for all $g$, $\Sigma(g)\neq\phi$. We fix $\hat{g}\in\Sigma(g)$ for every
$g$ and denote $\check{g}=\widehat{g^{-1}}$. We have $(g,\hat{g})\approx|g|$
uniformly in $g$. Perhaps the function $g\to\hat{g}$ can be chosen to be
measurable but this will not affect us. Since $(g,\hat{g})\approx|g|$ and
$(g,x)\leq|g|$ for any $x$ we get that $(x,\hat{g})\gtrsim
min\\{(x,g),(g,\hat{g})\\}\approx(x,g)$. On the other hand, we get that
$|g|\geq(g,x)\gtrsim min\\{(g,\hat{g}),(\hat{g},x)\\}\approx
min\\{|g|,(\hat{g},x)\\}$ so:

$(g,x)\approx min\\{|g|,(\hat{g},x)\\}$ (2)

######  Lemma 3.7.

Suppose $\mu$ is quasi-conformal of dimension $D$. The measure $\mu$ is
Ahlfors regular of dimension $D=\frac{h}{\varepsilon}$ with respect to the
metric $d_{\varepsilon}$, i.e for any $\xi\in\partial G$ and $\rho\leq
diam(\partial G)$

$\mu(B_{\rho}(\xi))\asymp\rho^{D}$

###### Proof.

Let $\xi,\rho$ be as above and let $\gamma:[0,\infty)\to G$ be a roughly
geodesic ray with $\gamma(0)=1,\gamma(\infty)=\xi$. We know that
$B_{\rho}(\xi)\subseteq\Sigma(\gamma(\frac{-1}{\epsilon}ln(\rho)))$. Taking
the measures of both sides we get by the shadow lemma that
$\mu(B_{\rho}(\xi))\prec\rho^{D}$.

For the other inequality suppose $\eta\in\Sigma(\gamma(t))$, then
$(\eta,\xi)\gtrsim min\\{(\eta,\gamma(t)),(\gamma(t),\xi)\\}\approx t$.
Therefore $d_{\varepsilon}(\xi,\eta)\asymp e^{-\varepsilon(\xi,\eta)}\prec
e^{-\varepsilon t}$, so there exists $C>0$ independent of $\xi,\rho$ such that
if $t\geq\frac{-1}{\varepsilon}ln(\rho)+C$ then
$d_{\varepsilon}(\xi,\eta)<\rho$. We conclude that
$\Sigma(\gamma(\frac{-1}{\varepsilon}ln(\rho)+C))\subseteq B_{\rho}(\xi)$.
Taking the measures of these sets and using lemma 3.4 we get that
$\rho^{D}\prec\mu(B_{\rho}(\xi))$. ∎

As a result, since $h\neq 0$, $\mu$ has no atoms. Because $\mu$ is Radon we
also see that the measure class $[\mu]$ is determined uniquely by the
assumption that $\mu$ is quasi-conformal of dimension $D$.

######  Lemma 3.8.

If $\mu$ is quasi-conformal of dimension $D$ then $\mu$ and the Hausdorff
measure $H^{D}$ of dimension $D$ on $\partial G$ are in the same measure class
and the corresponding Radon-Nikodym derivative is bounded and bounded away
from $0$.

###### Proof.

The proof is taken from [9, Corollary 2.5.10]. Let $A$ be a measurable set,
and $U_{i}$ a countable cover of $A$ by sets of diameters $\rho_{i}<\delta$.
Each $U_{i}$ is contained in a ball $B_{i}$ of radius $\rho_{i}$ so
$\mu(A)\prec\sum_{i}\rho_{i}^{D}$. Taking $\delta\to 0$ we get $\mu(A)\prec
H^{D}(A)$ uniformly in A.

For the other direction, let $A$ be measurable. The measures $\mu$ and $H^{D}$
are Radon so for every $\epsilon>0$ there exist compact $K$ and open $U$ with
$K\subset A\subset U$ and $H^{D}(U\backslash K),\mu(U\backslash K)<\epsilon$.
Since $K$ is compact there exists $\delta$ such that any ball centered at a
point in $K$ of radius less than $\delta$ is contained in $U$. Let $U_{i}$ be
a countable cover of $K$ with sets of diameters $\rho_{i}<\delta$, ordered
such that $\rho_{i}$ is non-increasing. We can assume without loss of
generality that $U_{i}\cap(K\backslash\bigcup_{j<i}U_{j})\neq\phi$. Under this
assumption each $U_{i}$ is contained in a ball $B_{i}$ of radius $\rho_{i}$
centered at a point in $K\backslash\bigcup_{j<i}U_{j}$. Therefore
$B_{i}\subset U$ and if $C_{i}$ denote the balls radius $\frac{\rho_{i}}{2}$
centered at the same point as $B_{i}$ then $C_{i}$ are disjoint. Thus
$\sum\rho_{i}^{D}\prec\mu(U)$ and taking $\delta$ to $0$ we get that
$H^{D}(K)\prec\mu(U)$ uniformly in $A,\epsilon$. Now taking $\epsilon$ to $0$
we get $H^{D}(A)\prec\mu(A)$ uniformly in $A$. ∎

Since the Hausdorff measure class is independent of $\mu$ there is a unique
measure class containing a quasi-conformal measure of dimension $D$. In
addition since $\frac{d\mu}{dH^{D}}\asymp 1$, $H^{D}$ is quasi-conformal of
dimension $D$. We now fix a quasi-conformal probability measure $\mu$ of
dimension $D$ and call it the Patterson Sullivan measure of $(G,d)$. As a
corollary we see that $\mu$ is ergodic. Indeed if, $E$ is a $G$ invariant
subset and $\mu(E)\neq 0$ then the restriction of $\mu$ to $E$ is quasi-
conformal of dimension $D$ and is thus equivalent to $\mu$ so E is co-null.

Finally we will need the following strengthening of the shadow lemma:

######  Lemma 3.9.

There exists a constant $0<C$ such that for any $g\in G$ and any $s<|g|-C$:

$\mu(\\{\xi\in\partial G|(g,\xi)>s\\})\asymp e^{-hs}$

###### Proof.

Let $s>0$ and set $\rho=e^{-\epsilon s}$. If $(\xi,g)>s$ then by estimate 2,
$(\hat{g},\xi)\gtrsim s$. Thus by estimate 1 there exists $L>0$, independent
of $g,s$, such that $\\{\xi\in\partial G|(g,\xi)>s\\}\subset
B_{L\rho}(\hat{g})$. Using Ahlfors regularity we get one inequality. In the
other direction, by estimate 2 there exists $C>0$, independent of $g,\xi$,
such that if $(\hat{g},\xi)>s+C$ then $(g,\xi)>min\\{s,|g|-C\\}$, so if
$s<|g|-C$ then $(g,\xi)>s$. Thus by estimate 1 there exists $L>0$, independent
of $g,s$, satisfying $B_{\frac{\rho}{L}}(\hat{g})\subset\\{\xi\in\partial
G|(g,\xi)>s\\}$. The second inequality follows again by Ahlfors regularity. ∎

##  4\. Some Growth Estimates

In this section we will use the shadow lemma to obtain growth estimates on the
group $G$. We generalize many of the results of Garncarek ([15, Section 4]) to
the locally compact case, many of the proofs are similar. The main difference
occurs in lemma 4.3 which requires the assumption of unimodularity which holds
trivially for discrete groups. We keep the notation of the previous section.

First we deduce the following corollary from lemma 3.5.

######  Theorem 4.1.

If $G$ is non-elementary and $\alpha$ is large enough,
$\lambda(A_{R}(\alpha))\asymp_{\alpha}\lambda(B_{R}(1))\asymp e^{hR}$.

###### Proof.

The proof is essentially a double counting argument. Suppose $\alpha$ is large
enough for lemma 3.5 to hold. Consider the set
$A=\\{(g,\xi)|\xi\in\Sigma(g)\\}$. Using Fubini’s theorem we see that on the
one hand by lemma 3.5:

$\lambda\times\mu(A)=\int\lambda(\\{g\in
A_{R}(\alpha)|\xi\in\Sigma(g)\\})d\mu(\xi)\asymp_{\alpha}\int 1d\mu(\xi)=1$

On the other hand by lemma 3.4:

$\lambda\times\mu(A)=\int_{A_{R}(\alpha)}\mu(\Sigma(g))d\lambda(g)\asymp_{\alpha}\int_{A_{R}(\alpha)}e^{-hR}d\lambda(g)=\lambda(A_{R}(\alpha))e^{-hR}$

Comparing the two we get the result for annuli. The result for balls follows
by covering the ball with annuli and using the formula for the sum of a
geometric series. ∎

We will need one more lemma estimating the size of certain subsets of annuli.

######  Lemma 4.2.

For any $s,R,\alpha>0$ and $\xi\in\partial G$:

$\lambda(\\{g\in A_{R}(\alpha)|(g,\xi)>s\\})\prec_{\alpha}e^{h(R-s)}$

###### Proof.

Let $\gamma$ be a roughly geodesic ray with $\gamma(0)=1,\gamma(\infty)=\xi$
and $g$ as above. Note that $(\gamma(s),\xi)\approx s$, so
$(g,\gamma(s))\gtrsim min\\{(g,\xi),(\xi,\gamma(s))\\}\approx s$, but
$|\gamma(s)|\approx s$ so $s\gtrsim(g,\gamma(s))$ and therefore
$(g,\gamma(s))\approx s$. Opening up the definition of the Gromov product and
plugging in $|g|\approx_{\alpha}R,|\gamma(s)|\approx s$ we get
$d(g,\gamma)\approx R-s$. Thus $\\{g\in A_{R}(\alpha)|(g,\xi)>s\\}$ is
contained in a ball of radius $r\approx R-s$ centred at $\gamma(s)$ . Applying
lemma 4.1 we obtain the result. ∎

The following lemma is one of the key geometric ideas in the proof of
irreducibility of boundary representations. It generalizes the fact that given
any two words $w_{1},w_{2}$ in a free group one can find a generator $x$ such
that the product $w_{1}xw_{2}$ has no cancellation. Interestingly the proof
requires the added assumption of unimodularity, which holds trivially in the
discrete case.

######  Lemma 4.3.

Suppose G is unimodular and non-elementary. There exists $\tau$ such that for
all $g,h\in G$, $\mu(\\{g^{\prime}\in
B_{\tau}(g)||g^{\prime}|+|h|-2\tau\leq|g^{\prime}h|\\})\asymp 1$.

###### Proof.

Let $g,h\in G$, and for every $s\in G$ fix a roughly geodesic segment
$\gamma_{s}:[0,|s|]\rightarrow G$ in G with
$\gamma_{s}(0)=1,\gamma_{s}(|s|)=s$. Let $g^{\prime}\in B_{\tau}(g)$ such that
$|g^{\prime}|+|h|-2\tau>|g^{\prime}h|$, this is equivalent to
$(g^{\prime}h,1)_{g^{\prime}}>\tau$. We get:

$(\gamma_{g^{\prime}}(|g^{\prime}|-\tau),g^{\prime}\gamma_{h}(\tau))_{g^{\prime}}\gtrsim
min\\{(\gamma_{g}^{\prime}(|g^{\prime}|-\tau),1)_{g^{\prime}},(1,g^{\prime}h)_{g^{\prime}},(g^{\prime}h,g^{\prime}\gamma_{h}(\tau))_{g^{\prime}}\\}\approx\tau$

Since
$d(g^{\prime},\gamma_{g^{\prime}}(|g^{\prime}|-\tau)),d(g^{\prime},g^{\prime}\gamma_{h}(\tau))\approx\tau$,
we conclude that
$d(\gamma_{g^{\prime}}(|g^{\prime}|-\tau),g^{\prime}\gamma_{h}(\tau))\approx
0$. Similarly we have that:

$(\gamma_{g}(|g^{\prime}|-\tau),\gamma_{g^{\prime}}(|g^{\prime}|-\tau))\gtrsim
min\\{(\gamma_{g}(|g^{\prime}|-\tau),g),(g,g^{\prime}),(g^{\prime},\gamma_{g^{\prime}}(|g^{\prime}|-\tau))\\}\gtrsim|g^{\prime}|-\tau$

So since
$|\gamma_{g^{\prime}}(|g^{\prime}|-\tau)|,|\gamma_{g}(|g^{\prime}|-\tau)|\approx|g^{\prime}|-\tau$
we conclude that
$d(\gamma_{g^{\prime}}(|g^{\prime}|-\tau),\gamma_{g}(|g^{\prime}|-\tau))\approx
0$.

Putting everything together we get that
$d(g^{\prime}\gamma_{h}(\tau),\gamma_{g}(|g^{\prime}|-\tau))\approx 0$. Notice
that $\gamma_{g}(|g^{\prime}|-\tau)\in\gamma_{g}([|g|-2\tau,|g|])$, so if
$|g^{\prime}|+|h|-2\tau>|g^{\prime}h|$, then $g^{\prime}\gamma_{h}(\tau)$ is
in a tubular neighborhood of constant distance (independent of $\tau$) from
the roughly geodesic segment $\gamma_{g}([|g|-2\tau,|g|])$. Denote this
neighborhood by $N(\gamma_{g}([|g|-2\tau,|g|]))$.

Consider the map $g^{\prime}\mapsto g^{\prime}\gamma_{h}(\tau)$. This map is
measure preserving since $G$ is unimodular. Because the measure of
$B_{\tau}(g)$ grows exponentially with $\tau$ and the measure of
$N(\gamma_{g}([|g|-2\tau,|g|]))$ grows linearly with $\tau$, for $\tau$ large
enough this map must send more than half of the measure of $B_{\tau}(g)$
outside of $N(\gamma_{g}([|g|-2\tau,|g|]))$. The elements not being sent to
$N(\gamma_{g}([|g|-2\tau,|g|]))$ satisfy
$|g^{\prime}|+|h|-2\tau\leq|g^{\prime}h|$. ∎

##  5\. Operators on $L^{2}(\mu)$

In this section we use the tools we have developed to construct operators in
the von Neuman algebra of the representation $\pi$. These operators will later
be used to show irreducibility of the representation. The results generalize
[15, Section 5] and many of the core ideas come from there, although some of
the constructions become more subtle and need to be changed.

###  5.1. Estimates on $\pi(g)$

To simplify calculations we assume without loss of generality that $D>1$.
Denote $P_{g}(\xi)=\sqrt{\frac{dg_{*}\mu}{d\mu}}(\xi)$,
$\widetilde{P}_{g}=\frac{P_{g}}{\|P_{g}\|_{1}}$ and
$\widetilde{\pi}(g)=\frac{\pi(g)}{\|P_{g}\|_{1}}$. Note that
$\|P_{g}\|_{1}=\langle\pi(g)1,1\rangle=\langle
1,\pi(g^{-1})1\rangle=\|P_{g^{-1}}\|_{1}$ so that
$\widetilde{\pi}(g)^{*}=\widetilde{\pi}(g^{-1})$. We will call the weak
operator closure of the image of the positive $L^{1}$ functions under $\pi$
the positive cone of $\pi$.

We will use the following lemma several times during calculations.

######  Lemma 5.1.

Let $(X,\mu)$ be a $\sigma$-finite measure space and $f$ a positive measurable
function on $X$ then:

$\int_{X}f(\xi)d\mu=\int\limits_{0}^{\infty}\mu(\\{\xi|f(\xi)>t\\})dt$

###### Proof.

The proof follows from Fubini’s theorem:

$\int_{X}f(\xi)d\mu=\int_{X}\int\limits_{0}^{\infty}\mathbbm{1}_{\\{(t,\xi)|t<f(\xi)\\}}dtd\mu=\int\limits_{0}^{\infty}\int_{X}\mathbbm{1}_{\\{(t,\xi)|t<f(\xi)\\}}d\mu
dt=\int\limits_{0}^{\infty}\mu(\\{\xi|f(\xi)>t\\})dt$

∎

######  Lemma 5.2.

The following estimates are satisfied uniformly in $g,\xi$:

$\|P_{g}\|_{1}\asymp e^{-\frac{h|g|}{2}}(1+|g|)$
$\widetilde{P}_{g}\asymp\frac{e^{h(g,\xi)}}{1+|g|}$

###### Proof.

By definition $P_{g}(\xi)\asymp e^{h((g,\xi)-\frac{|g|}{2})}$. Let $C$ be the
constant from lemma 3.9. If $|g|<C$ then $\|P_{g}\|_{1}$ is bounded by a
constant. Assume $|g|\geq C$. By lemma 3.9 $\mu(\\{\xi\in\partial
G|(g,\xi)>\frac{ln(t)}{h}\\})\asymp\frac{1}{t}$ for $t<|g|-C$. we get:

$\displaystyle e^{\frac{|g|}{2}}\int_{\partial G}P_{g}(\xi)d\mu$
$\displaystyle\asymp\int_{\partial G}e^{h(g,\xi)}d\mu$
$\displaystyle=\int_{0}^{\infty}\mu(\\{\xi\in\partial
G|(g,\xi)>\frac{ln(t)}{h}\\})dt$ $\displaystyle\asymp
1+\int_{1}^{e^{h(|g|-C)}}\mu(\\{\xi\in\partial
G|(g,\xi)>\frac{ln(t)}{h}\\})dt$ $\displaystyle\asymp
1+\int_{1}^{e^{h(|g|-C)}}\frac{dt}{t}$ $\displaystyle=1+h(|g|-C)$
$\displaystyle\asymp 1+|g|$

Which gives the first estimate. The second estimate now follows from the first
and from $P_{g}(\xi)\asymp e^{h((g,\xi)-\frac{|g|}{2})}$.

∎

######  Lemma 5.3.

For every $R$,$\alpha$ and for almost every $\xi\in\partial G$

$\int_{A_{R}(\alpha)}\widetilde{P}_{g}(\xi)d\lambda(g)\prec_{\alpha}e^{hR}$

###### Proof.

Using theorem 4.1, lemma 4.2 and the previous lemma we see that:

$\displaystyle(1+R)\int_{A_{R}(\alpha)}\widetilde{P}_{g}(\xi)d\lambda$
$\displaystyle\asymp_{\alpha}\int_{A_{R}(\alpha)}e^{h(g,\xi)}d\lambda$
$\displaystyle=\int_{0}^{\infty}\lambda\\{g\in
A_{R}(\alpha)|(g,\xi)>\frac{ln(t)}{h}\\}dt$
$\displaystyle\asymp_{\alpha}e^{hR}+\int_{1}^{e^{h(R+\alpha)}}\lambda\\{g\in
A_{R}(\alpha)|(g,\xi)>\frac{ln(t)}{h}\\}dt$
$\displaystyle\prec_{\alpha}e^{hR}+\int_{1}^{e^{h(R+\alpha)}}e^{h(R-\frac{ln(t)}{h})}dt$
$\displaystyle=e^{hR}(1+R+\alpha)$

Giving the required estimate. ∎

We now obtain estimates on matrix coefficients of the representation $\pi$.
Denote the space of $d_{\varepsilon}$-Lipschitz functions on $\partial G$ by
$Lip(\partial G)$, and for $\phi\in Lip(\partial G)$ denote the Lipschitz
constant by $L(\phi)$. $Lip(\partial G)$ is dense in $L^{2}(\mu)$. To see this
notice that by the Lebesgue differentiation theorem, which holds for Ahlfors
regular measures by [16, Theorem 1.8], every non-zero $L^{2}$ function has
non-zero inner product with the characteristic function of some ball. Thus
characteristic functions of balls span a dense subspace of $L^{2}(\mu)$. Since
these can be approximated by Lipschitz functions, $Lip(\partial G)$ is dense
in $L^{2}(\mu)$.

######  Lemma 5.4.

Let $g\in G$. For any $\phi,\psi\in Lip(\partial G)$ we have:

$|\langle\widetilde{\pi}(g)\phi,\psi\rangle-\phi(\check{g})\overline{\psi(\hat{g})}|\prec\frac{L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty}}{(1+|g|)^{\frac{1}{D}}}$

###### Proof.

By the choice of normalization $\langle\widetilde{\pi}(g)1,1\rangle=1$.
Therefore:

$\displaystyle|\langle\widetilde{\pi}(g)\phi,\psi\rangle-\phi(\check{g})\overline{\psi(\hat{g})}|$
$\displaystyle=$
$\displaystyle|\langle\widetilde{\pi}(g)\phi,\psi\rangle-\langle\widetilde{\pi}(g)\phi(\check{g}),{\psi(\hat{g})}\rangle|$
$\displaystyle\leq$
$\displaystyle|\langle\widetilde{\pi}(g)\phi,\psi-\psi(\hat{g})\rangle|+|\langle\widetilde{\pi}(g)(\phi-\phi(\check{g})),\psi(\hat{g})\rangle|$
$\displaystyle=$
$\displaystyle|\langle\widetilde{\pi}(g)\phi,\psi-\psi(\hat{g})\rangle|+|\langle\phi-\phi(\check{g}),\widetilde{\pi}(g^{-1})\psi(\hat{g})\rangle|$

We estimate the first term, the second term can be estimated similarly after
replacing $g$ by $g^{-1}$. Let $\rho>0$ and consider the ball
$B_{\rho}(\hat{g})$. We will decompose the integral over $B_{\rho}(\hat{g})$
and the complement.

$\displaystyle|\langle\widetilde{\pi}(g)\phi,\psi-\psi(\hat{g})\rangle|$
$\displaystyle\leq$ $\displaystyle\int_{\partial
G}\widetilde{P}_{g}(\xi)|\phi(g^{-1}\xi)||\psi(\xi)-\psi(\hat{g})|d\mu(\xi)$
$\displaystyle=$
$\displaystyle\int_{B_{\rho}(\hat{g})}\widetilde{P}_{g}(\xi)|\phi(g^{-1}\xi)||\psi(\xi)-\psi(\hat{g})|d\mu(\xi)+$
$\displaystyle\int_{\partial G\backslash
B_{\rho}(\hat{g})}\widetilde{P}_{g}(\xi)|\phi(g^{-1}\xi)||\psi(\xi)-\psi(\hat{g})|d\mu(\xi)$

For $\xi\in B_{\rho}(\hat{g})$, $|\psi(\xi)-\psi(\hat{g})|\leq L(\psi)\rho$ so
the first integral is bounded by $\|\phi\|_{\infty}L(\psi)\rho$. In order to
estimate the second integral recall that for any $\xi$,
$(\hat{g},\xi)\geq(g,\xi)$, so that
$\widetilde{P}_{g}\asymp\frac{e^{h(g,\xi)}}{1+|g|}\prec\frac{e^{h(\hat{g},\xi)}}{1+|g|}\asymp\frac{d_{\varepsilon}(\hat{g},\xi)^{-D}}{1+|g|}$.
Thus (using the assumption $D>1$):

$\displaystyle\int_{\partial G\backslash
B_{\rho}(\hat{g})}\widetilde{P}_{g}(\xi)|\phi(g^{-1}\xi)||\psi(\xi)-\psi(\hat{g})|d\mu(\xi)$
$\displaystyle\prec$ $\displaystyle\int_{\partial G\backslash
B_{\rho}(\hat{g})}\frac{\|\phi\|_{\infty}L(\psi)d_{\varepsilon}(\hat{g},\xi)^{1-D}}{1+|g|}d\mu(\xi)$
$\displaystyle\leq$ $\displaystyle\int_{\partial G\backslash
B_{\rho}(\hat{g})}\frac{\|\phi\|_{\infty}L(\psi)\rho^{1-D}}{1+|g|}d\mu(\xi)\leq\frac{\|\phi\|_{\infty}L(\psi)\rho^{1-D}}{1+|g|}$

All in all we get a bound on the first term proportional to
$\|\phi\|_{\infty}L(\psi)(\rho+\frac{\rho^{1-D}}{1+|g|})$ and similarly a
bound on the second term proportional to
$\|\psi\|_{\infty}L(\phi)(\rho+\frac{\rho^{1-D}}{1+|g|})$. Finally:

$|\langle\widetilde{\pi}(g)\phi,\psi\rangle-\phi(\check{g})\overline{\psi(\hat{g})}|\prec(\|\phi\|_{\infty}L(\psi)+\|\psi\|_{\infty}L(\phi))(\rho+\frac{\rho^{1-D}}{1+|g|})$

One can now differentiate with respect to $\rho$ in order to find the best
bound. To obtain the claim it is enough to take $\rho=(1+|g|)^{-\frac{1}{D}}$.

∎

###  5.2. Kernel operators

Given a function $K\in L^{\infty}(\partial^{2}G)$ we can define a bounded
operator $T_{K}$ on $L^{2}(\mu)$ by $T_{K}f(\xi)=\int
K(\xi,\eta)f(\eta)d\mu(\eta)$. These operators are called kernel operators.
The key theorem is that kernel operators with non-negative kernel $K$ are in
the positive cone of $\pi$. This will give us a rich enough family of
operators to show the representation is irreducible.

Assume now that $G$ is unimodular so lemma 4.3 holds and fix and a positive
kernal $K\geq 0$ in $L^{\infty}(\partial^{2}G)$. Given $R$, consider a family
$F\subset A_{R}(\alpha)$ as in lemma 3.6, for some fixed large enough
$\alpha$. Fix $\tau$ as in lemma 4.3 and denote $B=B_{\tau}(1)$. We partition
$\partial^{2}G$ as follows, choose a linear order on $F^{2}$, the sets
$\Sigma(g)\times\Sigma(h)$ cover $\partial^{2}G$ for $g,h\in F$. Using the
order on $F^{2}$ we define
$V_{g,h}=\Sigma(g)\times\Sigma(h)\backslash\bigcup_{(s,t)<(g,h)}\Sigma(s)\times\Sigma(t)$.
After discarding empty sets $\\{V_{g,h}|(g,h)\in F^{2}\\}$ is a partition of
$\partial^{2}G$. Denote $U_{g,h}=\\{k\in gBh^{-1}||k|>|g|+|h|-3\tau\\}$, by
lemma 4.3 and since $G$ is unimodular $\lambda(U_{g,h})\asymp 1$. Note that
$U_{g,h}\subseteq A_{2R}(2\alpha+3\tau)$. Define:

$w_{g,h}=\frac{\int_{V_{g,h}}Kd\mu^{2}}{\lambda(U_{g,h})}$

Denote also $w(k)=\sum_{F^{2}}w_{g,h}\mathbbm{1}_{U_{g,h}}(k)$ and
$S_{R,g,h}=\int_{U_{g,h}}\widetilde{\pi}(k)d\lambda(k)$. Finaly define:

$S_{R}=\int_{G}w(k)\widetilde{\pi}(k)d\lambda(k)=\sum_{F^{2}}w_{g,h}\int_{U_{g,h}}\widetilde{\pi}(k)d\lambda(k)=\sum_{F^{2}}w_{g,h}S_{R,g,h}$

We now prove that if $G$ is a non-elementary unimodular locally compact
hyperbolic group then $S_{R}\underset{R\to\infty}{\longrightarrow}T_{K}$ in
the weak operator topology. This follows easily from the next two lemmas.

######  Lemma 5.5.

If $G$ is a non-elementary unimodular locally compact hyperbolic group and
$\phi,\psi\in Lip(\partial G)$ then $\langle
S_{R}\phi,\psi\rangle\underset{R\to\infty}{\longrightarrow}\langle
T_{K}\phi,\psi\rangle$.

###### Proof.

If $k\in U_{g,h}$ then $|k|\approx|g|+|h|$, since $d(g,k)\lesssim|h|$ we get
that $(g,k)\approx|g|$. Thus we have $(\hat{g},\hat{k})\gtrsim
min\\{(\hat{g},g)(g,k)(k,\hat{k})\\}\approx|g|\approx R$ so
$d_{\varepsilon}(\hat{g},\hat{k})\prec e^{-\varepsilon R}$. Similarly
$d_{\varepsilon}(\hat{h},\check{k})\prec e^{-\varepsilon R}$. Therefore
$|\phi(\hat{h})\overline{\psi(\hat{g})}-\phi(\check{k})\overline{\psi(\hat{k})}|\prec(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})e^{-\varepsilon
R}$, so by lemma 5.4:

$\displaystyle|\langle\widetilde{\pi}(k)\phi,\psi\rangle-\phi(\hat{h})\overline{\psi(\hat{g})}|$
$\displaystyle<$
$\displaystyle|\langle\widetilde{\pi}(k)\phi,\psi\rangle-\phi(\check{k})\overline{\psi(\hat{k})}|+|\phi(\check{k})\overline{\psi(\hat{k})}-\phi(\hat{h})\overline{\psi(\hat{g})}|$
$\displaystyle\prec$
$\displaystyle(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})\left(\frac{1}{(1+R)^{\frac{1}{D}}}+e^{-\varepsilon
R}\right)$

As a result we get:

$\left|\frac{1}{\lambda(U_{g,h})}\left\langle
S_{R,g,h}\phi,\psi\right\rangle-\phi(\hat{h})\overline{\psi(\hat{g})}\right|\prec(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})\left(\frac{1}{(1+R)^{\frac{1}{D}}}+e^{-\varepsilon
R}\right)$

In addition, if $\xi,\eta\in V_{g,h}$ then
$d_{\varepsilon}(\xi,\hat{g}),d_{\varepsilon}(\eta,\hat{h})\prec
e^{-\varepsilon R}$ so
$|\phi(\hat{h})\overline{\psi(\hat{g})}-\phi(\eta)\overline{\psi(\xi)}|\prec(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})e^{-\varepsilon
R}$ and thus:

$\displaystyle\int_{V_{g,h}}K(\xi,\eta)|\phi(\eta)\overline{\psi(\xi)}-\phi(\hat{h})\overline{\psi(\hat{g})}|d\mu^{2}(\eta,\xi)$
$\displaystyle\prec$
$\displaystyle\int_{V_{g,h}}K(\xi,\eta)d\mu^{2}(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})e^{-\varepsilon
R}$

Finally, using the previous two estimates:

$\displaystyle|\langle S_{R}\phi,\psi\rangle-\langle T_{K}\phi,\psi\rangle|$
$\displaystyle\leq$ $\displaystyle\sum_{F^{2}}\left|w_{g,h}\langle
S_{R,g,h}\phi,\psi\rangle-\int_{V_{g,h}}K(\xi,\eta)\phi(\eta)\overline{\psi(\xi)}d\mu^{2}(\xi,\eta)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{F^{2}}\left(\int_{V_{g,h}}Kd\mu^{2}\right)\left|\frac{1}{\lambda(U_{g,h})}\langle
S_{R,g,h}\phi,\psi\rangle-\phi(\hat{h})\overline{\psi(\hat{g})}\right|$
$\displaystyle+$
$\displaystyle\sum_{F^{2}}\int_{V_{g,h}}K(\xi,\eta)|\phi(\hat{h})\overline{\psi(\hat{g})}-\phi(\eta)\overline{\psi(\xi)}|d\mu^{2}(\eta,\xi)$
$\displaystyle\prec$
$\displaystyle\sum_{F^{2}}\left[\left(\int_{V_{g,h}}Kd\mu^{2}\right)(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})\left(\frac{1}{(1+R)^{\frac{1}{D}}}+2e^{-\varepsilon
R}\right)\right]$ $\displaystyle=$
$\displaystyle\|K\|_{1}(L(\phi)\|\psi\|_{\infty}+L(\psi)\|\phi\|_{\infty})\left(\frac{1}{(1+R)^{\frac{1}{D}}}+2e^{-\varepsilon
R}\right)$

The right hand side tends to $0$ as $R\to\infty$ as needed.

∎

######  Lemma 5.6.

If $G$ is a non-elementary unimodular locally compact hyperbolic group then
the norms $\|S_{R}\|_{op}$ are bounded uniformly in R.

###### Proof.

We first show $w(x)\prec e^{-2hR}$. For every $(g,h)\in F^{2}$,
$V_{g,h}\subseteq\Sigma(g)\times\Sigma(h)$, so by lemma 3.4
$\int_{V_{g,h}}Kd\mu^{2}\prec\|K\|_{\infty}e^{-2hR}\prec e^{-2hR}$. Since
$\lambda(U_{g,h})\asymp 1$ we get that $w_{g,h}\prec e^{-2hR}$. Now all we
need to show is that the $U_{g,h}$ are disjoint. Indeed if $U_{g,h}\cap
U_{g^{\prime},h^{\prime}}\neq\phi$ there exist $s\in B_{\tau}(g),s^{\prime}\in
B_{\tau}(g^{\prime})$ such that
$|sh^{-1}|\geq|s|+|h|-4\tau,|s^{\prime}h^{\prime-1}|\geq|s^{\prime}|+|h^{\prime}|-4\tau$
and $sh^{-1}=s^{\prime}h^{\prime-1}$. $\tau$ is a fixed constant so
$(sh^{-1},s)\gtrsim|s|\approx
R,(s^{\prime}h^{\prime-1},s^{\prime})\gtrsim|s^{\prime}|\approx R$. Since
$|g|,|g^{\prime}|,|s|,|s^{\prime}|\approx R$ and
$d(g,s),d(g^{\prime},s^{\prime})\approx 0$, we get
$(g,s),(g^{\prime},s^{\prime})\approx R$. Thus using
$sh^{-1}=s^{\prime}h^{\prime-1}$, $(g,g^{\prime})\gtrsim
min\\{(g,s),(s,sh^{-1})(s^{\prime}h^{\prime-1},s^{\prime}),(s^{\prime},g^{\prime})\\}\approx
R$. Since $|g|,|g^{\prime}|\approx R$ we get that $d(g,g^{\prime})\approx 0$.
Taking inverses we similarly see that $d(h,h^{\prime})\approx 0$. As a result
there exists a constant $C$ depending only on $(G,d)$ such that if
$U_{g,h}\cap U_{g^{\prime},h^{\prime}}\neq\phi$ then
$d(g,g^{\prime}),d(h,h^{\prime})<C$. By the definition of $F$ (and our
assumptions on the choice of $\sigma$ defining $\Sigma(g)=\Sigma(g,\sigma)$)
we know that for this specific $C$, lemma 3.6 holds and thus since
$d(g,g^{\prime}),d(h,h^{\prime})<C$ we conclude that $g=g^{\prime}$ and
$h=h^{\prime}$ as needed. Note that there is no circular reasoning in the
proof since the constant $C$ which we obtained depends only on $(G,d)$ so our
choice of the parameter $\sigma$ is independent of $R$.

Using $w(x)\prec e^{-2hR}$ and the fact that $w(k)$ is supported in
$A_{2R}(2\alpha+3\tau)$ it follows from lemma 5.3 that
$\|S_{R}\mathbbm{1}_{\partial G}\|_{\infty}\prec 1$ because:

$S_{R}\mathbbm{1}_{\partial G}=\int
w(k)\widetilde{\pi}(k)\mathbbm{1}_{\partial G}d\lambda(k)=\int
w(k)\widetilde{P}_{g}d\lambda(k)\prec\int_{A_{2R}(2\alpha+3\tau)}e^{-2hR}\widetilde{P}_{g}d\lambda(k)\prec
1$

Since $G$ is unimodular,

$S_{R}^{*}=\int w(g)\widetilde{\pi}(g^{-1})d\lambda(g)=\int
w(g^{-1})\pi(g)d\lambda(g)$

(otherwise we would need right Haar measure). Using this formula one can now
similarly show that $\|S_{R}^{*}\mathbbm{1}_{\partial G}\|_{\infty}\prec 1$.

Let $\phi,\psi\in L^{4}(\mu)$ and consider the measure
$d\alpha=w(g)d\lambda(g)$. Using Cauchy-Shwartz and our estimates on
$\|S_{R}\mathbbm{1}_{\partial G}\|_{\infty},\|S_{R}^{*}\mathbbm{1}_{\partial
G}\|_{\infty}$ we have:

$\displaystyle|\langle S_{R}\phi,\psi\rangle|^{2}=\left|\int_{G\times\partial
G}\widetilde{P}_{g}(\xi)\phi(g^{-1}\xi)\overline{\psi(\xi)}d\alpha\times\mu(g,\xi)\right|^{2}$
$\displaystyle\leq$ $\displaystyle\left|\int_{G\times\partial
G}\widetilde{P}_{g}(\xi)|\phi(g^{-1}\xi)|^{2}d\alpha\times\mu(g,\xi)\right|\left|\int_{G\times\partial
G}\widetilde{P}_{g}(\xi)|\psi(\xi)|^{2}d\alpha\times\mu(g,\xi)\right|$
$\displaystyle=$ $\displaystyle|\langle S_{R}|\phi|^{2},\mathbbm{1}_{\partial
G}\rangle||\langle S_{R}\mathbbm{1}_{\partial
G},|\psi|^{2}\rangle|=|\langle|\phi|^{2},S_{R}^{*}\mathbbm{1}_{\partial
G}\rangle||\langle S_{R}\mathbbm{1}_{\partial G},|\psi|^{2}\rangle|$
$\displaystyle\leq$ $\displaystyle\|S_{R}^{*}\mathbbm{1}_{\partial
G}\|_{\infty}\|S_{R}\mathbbm{1}_{\partial
G}\|_{\infty}\|\phi^{2}\|_{1}\|\psi^{2}\|_{1}\prec\|\phi\|_{2}\|\psi\|_{2}$

The claim now follows by density of $L^{4}(\mu)$ in $L^{2}(\mu)$. ∎

######  Theorem 5.7.

Let $G$ be a unimodular, non-elementary, locally compact hyperbolic group. Let
$d\in\mathcal{D}(G)$ and let $\mu$, $\pi$ be the corresponding measure and
boundary representation. The kernel operators with non-negative kernels are in
the positive cone of $\pi$.

###### Proof.

Since the operators $S_{R}$ have uniformly bounded norms, in order to check
convergence in the weak operator topology it is enough to check that $\langle
S_{R}\phi,\psi\rangle\underset{R\to\infty}{\longrightarrow}\langle
T_{K}\phi,\psi\rangle$ for $\phi,\psi$ in a dense subset of $L^{2}(\mu)$. When
$K\geq 0$ we saw that this convergence holds for $\phi,\psi\in Lip(\partial
G)$. ∎

We now prove one more lemma we will need later.

######  Lemma 5.8.

For any Borel set $E\subseteq\partial G$ the projection $P_{E}$ on to
$L^{2}(E)$ is in the positive cone of $\pi$.

###### Proof.

Consider the kernels
$K_{\rho}(\xi,\eta)=\frac{1}{\mu(B_{\rho}(\xi))}\mathbbm{1}_{\\{(\xi,\eta)|\xi\in
E,d(\xi,\eta)<\rho\\}}$ with corresponding kernel operators $T_{\rho}$ which
are in the positive cone. We claim that $T_{\rho}\to P_{E}$ in the weak
operator topology. As before we check convergence on Lipschitz functions
first. Let $\phi,\psi\in Lip(\partial G)$ and $\xi\in\partial G$

$\displaystyle\left|\int
K_{\rho}(\xi,\eta)\phi(\eta)d\mu(\eta)-P_{E}\phi(\xi)\right|$ $\displaystyle=$
$\displaystyle\frac{\mathbbm{1}_{E}(\xi)}{\mu(B_{\rho}(\xi))}\left|\int_{B_{\rho}(\xi)}(\phi(\eta)-\phi(\xi))d\mu(\eta)\right|$
$\displaystyle\leq$
$\displaystyle\frac{\mathbbm{1}_{E}(\xi)}{\mu(B_{\rho}(\xi))}\int_{B_{\rho}(\xi)}L(\phi)\rho
d\mu\leq L(\phi)\rho$

Using this estimate we see that:

$|\langle(T_{\rho}-P_{E})\phi,\psi\rangle|\leq\|\psi\|_{\infty}L(\phi)\rho\underset{\rho\to
0}{\longrightarrow}0$

Now since $Lip(\partial G)$ is dense in $L^{2}(\partial G)$ all that remains
is to check that $\|T_{\rho}\|_{op}$ are bounded independently of $\rho$. For
any non negative kernel $K$ it follows from the Cauchy-Shwartz inequality on
$(\partial G^{2},\mu^{2})$ that for any $\phi,\psi\in L^{2}(\mu)$

$\langle
T_{K}\phi,\psi\rangle\leq\|\sqrt{K(\xi,\eta)}\phi(\eta)\|_{2}\|\sqrt{K(\xi,\eta)}\overline{\psi(\xi)}\|_{2}\leq\left\|\int
K(\xi,\eta)d\mu(\xi)\right\|_{\infty}^{\frac{1}{2}}\left\|\int
K(\xi,\eta)d\mu(\eta)\right\|_{\infty}^{\frac{1}{2}}\|\phi\|_{2}\|\psi\|_{2}$

But for $K=K_{\rho}$ it is easily seen that:

$\int K_{\rho}(\xi,\eta)d\mu(\eta),\int K_{\rho}(\xi,\eta)d\mu(\xi)\leq 1$

So $\|T_{\rho}\|\leq 1$ as needed.

∎

##  6\. Irreducibility of Boundary Representations

We now prove irreducibility of the representation $\pi$.

######  Lemma 6.1.

Let $\pi$ be a representation of $G$, if the weak operator closure of
$\pi(L^{1}(G))$ contains a projection on a cyclic vector then $\pi$ is
irreducible.

###### Proof.

We show that the commutant of $\pi$ is trivial, the result then follows by
Schur’s lemma. Let $P$ be a projection on to a cyclic vector $v$. If $P$ is in
the weak operator closure of $\pi(L^{1}(G))$ then so is $\pi(g)P\pi(g)^{-1}$,
which is the projection on $\pi(g)v$. So if $T$ is an operator commuting with
$\pi(L^{1}(G))$ then $\pi(g)v$ is an eigenvector of $T$ for every $g\in G$.
Since the vectors $\pi(g)v$ span a dense subspace we conclude that $T$ is a
scalar operator. ∎

If G is a unimodular non-elementary locally compact hyperbolic group and $\pi$
is the boundary representation corresponding to some $d\in\mathcal{D}(G)$ then
all the kernel operators are in the weak operator closure of $\pi(L^{1}(G))$.
Let $\phi\in L^{\infty}(\mu)$ and consider the kernel operator $T_{K}$
corresponding to the kernel $K(\xi,\eta)=\phi(\xi)$. Note that
$T_{K}(\mathbbm{1}_{\partial G})=\phi$. Therefore since $L^{\infty}(\mu)$ is
dense in $L^{2}(\mu)$, $\mathbbm{1}_{\partial G}$ is cyclic. Taking
$\phi=\mathbbm{1}_{\partial G}$ we see that the projection on
$\mathbbm{1}_{\partial G}$ is in the weak operator closure of $\pi(L^{1}(G))$.
Thus we get by the lemma:

######  Theorem 6.2.

Let $G$ be a unimodular, non-elementary, hyperbolic locally compact group. Let
$d\in\mathcal{D}(G)$ and let $\mu,\pi$ be the the corresponding Patterson-
Sullivan measure and boundary representation then $\pi$ is irreducible.

Finishing the proof of theorem 1.1.

We remark that unimodularity of $G$ only entered the proof of irreducibility
in two places: lemma 4.3 and lemma 5.6, although the entire construction of
the previous section would not work without these two lemmas.

##  7\. Rough Equivalence of Metrics

This section is dedicated to the proof that two metrics
$d_{1},d_{2}\in\mathcal{D}(G)$ are roughly similar if and only if the
corresponding boundary representations are unitarily equivalent. The proof
will use the fact that the action of $G$ on $\partial^{2}G$ is erogodic which
we will show later (theorem 10.5), independently of this section.

We fix throughout the section metrics $d_{1},d_{2}\in\mathcal{D}(G)$. We
denote objects corresponding to $d_{i}$ by a subscript $i$. For example the
boundary representations will be $\pi_{i}$ and the visual metrics will have
parameters $\epsilon_{i}$ and be denoted $d_{\epsilon_{i}}$. From this point
forward we add the assumption that $G$ is second countable to the standing
assumption that $G$ is non elementary and unimodular.

######  Lemma 7.1.

If the representations $\pi_{1}$ and $\pi_{2}$ are unitarily equivalent then
there exists an almost everywhere defined measure class preserving isomorphism
$F:\partial G\to\partial G$ conjugating $\pi_{1}$ to $\pi_{2}$.

###### Proof.

Denote the ($G$ equivariant) unitary isomorphism $T:L^{2}(\mu_{1})\to
L^{2}(\mu_{1})$. $T$ induces an isomorphism of the von Neumann algebras of
bounded operators on these Hilbert spaces $\hat{T}:B(L^{2}(\mu_{1}))\to
B(L^{2}(\mu_{2}))$ by $\hat{T}(A)=TAT^{-1}$. We claim that $\hat{T}$ restricts
to an isomorphism between the von Neumann algebras $L^{\infty}(\mu_{i})$
considered as multiplication operators inside $B(L^{2}(\mu_{i}))$.

The projections in $L^{\infty}(\mu_{i})$ can be characterized amongst the
projections of $B(L^{2}(\mu_{i}))$ as those $P$ such that both $P$ and $I-P$
preserve the partial order on $L^{2}(\mu_{i})$. Obviously all projections in
$L^{\infty}(\mu_{i})$ satisfy this. In the other direction if $P$ and $I-P$
both preserve the partial order on $L^{2}(\mu_{i})$ then
$P\mathbbm{1}_{\partial G}$,$(I-P)\mathbbm{1}_{\partial G}$ are positive
orthogonal functions, so they must have disjoint supports. Since
$P\mathbbm{1}_{\partial G}+(I-P)\mathbbm{1}_{\partial G}=\mathbbm{1}_{\partial
G}$ we conclude that $P\mathbbm{1}_{\partial G}$ is the indicator function of
some set $E$. If $\phi\in L^{\infty}(\mu_{i})$ is positive then $P\phi$ is
supported on $E$ since $0\leq P\phi<P\|\phi\|_{\infty}\mathbbm{1}_{\partial
G}=\|\phi\|_{\infty}\mathbbm{1}_{E}$. Using linearity and density of
$L^{\infty}(\mu_{i})$ in $L^{2}(\mu_{i})$ we deduce that $Im(P)\subseteq
L^{2}(E)$. Similarly $Im(I-P)\subseteq L^{2}(E^{c})$, so since $P,I-P$ are
projections which sum up to $I$ we deduce that $Im(P)=L^{2}(E)$ as needed.

By lemma 5.8 we know that all the projections in $L^{\infty}(\mu_{1})$ are in
the positive cone of $\pi_{1}$. Using the $G$-equivariance we see that
$\hat{T}$ takes the positive cone of $\pi_{1}$ in to the positive cone of
$\pi_{2}$. Since elements of the positive cone of $\pi_{2}$ preserve the
partial order on $L^{2}(\mu_{2})$ we conclude that $\hat{T}$ takes projections
in $L^{\infty}(\mu_{1})$ to projections in $L^{\infty}(\mu_{2})$. Finally
because the projections generate $L^{\infty}(\mu_{1})$ we conclude that
$\hat{T}(L^{\infty}(\mu_{1}))\subseteq L^{\infty}(\mu_{2})$. Applying the same
line of reasoning to $\hat{T}^{-1}$ one gets the opposite inclusion showing
that $\hat{T}$ restricts to a $G$-equivariant isomorphism of von Neumann
algebras $\hat{T}:L^{\infty}(\mu_{1})\to L^{\infty}(\mu_{2})$

Since maps between commutative von Neumann algebras correspond bijectively to
almost everywhere defined maps between the spectra we conclude that there
exists a $G$-equivariant isomorphism $F:(\partial G,\mu_{1})\to(\partial
G,\mu_{2})$.

∎

Our strategy now will be first to show that $F$ agrees almost everywhere with
a continuous map and then to show that $F$ is the identity so that $\mu_{1}$
and $\mu_{2}$ are actually in the same measure class. In order to do this we
will take a slight detour.

Denote by $\partial^{2}G$ the space of distinct pairs of elements in $\partial
G$. It will be slightly more convenient for us to consider only distinct pairs
but since the measure $\mu$ is purely non-atomic this has no measure theoretic
significance.

######  Theorem 7.2.

For any Patterson-Sullivan measure $\mu$, the measure space
$(\partial^{2}G,\mu^{2})$ contains an invariant (infinite) measure $m$
satisfying:

$dm(\xi,\eta)\asymp e^{2h(\xi,\eta)}d\mu^{2}(\xi,\eta)\asymp
d_{\epsilon}(\xi,\eta)^{-2D}d\mu^{2}(\xi,\eta)$

To prove this theorem we will need two lemmas.

######  Lemma 7.3.

Let $G$ be a second countable locally compact group acting continuously on a
locally compact space $X$. If $G$ has a dense subgroup $H$ preserving a
(possibly infinite) Radon measure $m$ then $G$ preserves $m$.

###### Proof.

Let $f:X\to\mathbb{R}$ be a continuous compactly supported function. Let $g\in
G$ and $g_{n}\in H$ such that $g_{n}\to g$. If $U$ is some compact
neighborhood of $g$ then the functions $f(g^{-1}x)$ are all supported on some
compact subset of $X$. Since $g_{n}$ preserve $m$ we see by dominated
convergence that:

$\int f(x)dg_{*}m=\int f(g^{-1}x)dm=\lim_{n\to\infty}\int
f(g_{n}^{-1}x)dm=\lim_{n\to\infty}\int f(x)d(g_{n})_{*}m=\int f(x)dm$

Radon measures are defined by their values on continuous compactly supported
functions so since $f$ is arbitrary we conclude that $g_{*}m=m$ as needed. ∎

######  Lemma 7.4.

Let $G$ a locally compact second countable group acting continuously on a
locally compact space $X$ and preserving the measure class of a Radon measure
$\nu$ such that $\frac{dg_{*}\nu}{d\nu}\asymp 1$ uniformly in $g$. There
exists an invariant measure $m\in[\nu]$ satisfying $\frac{dm}{d\nu}\asymp 1$.

###### Proof.

We start with the case of a countable group. Consider the function:

$\alpha(x)=\sup_{h\in G}\frac{dh_{*}\nu}{d\nu}$

Since $G$ is countable $\alpha$ is measurable and well defined almost
everywhere. By the chain rule for Radon Nikodym derivatives:

$\frac{d(gh)_{*}\nu}{d\nu}(x)=\frac{dh_{*}\nu}{d\nu}(g^{-1}x)\frac{dg_{*}\nu}{d\nu}(x)$

Taking the supremum over $h\in G$ we get that for every $g\in G$:

$\frac{dg_{*}\nu}{d\nu}(x)=\frac{\alpha(x)}{\alpha(g^{-1}x)}$

If we define a measure $m$ by $dm=\alpha(x)d\nu$ then for any $g\in G$:

$\frac{dg_{*}m}{dm}(x)=\frac{dg_{*}m}{dg_{*}\nu}(x)\frac{dg_{*}\nu}{d\nu}(x)\frac{d\nu}{dm}(x)=\frac{\alpha(g^{-1}x)}{\alpha(x)}\frac{dg_{*}\nu}{d\nu}(x)=1$

So $m$ is invariant under $G$. By assumption $\alpha(x)\asymp 1$.

If now we assume that $G$ is locally compact and second countable then then
$G$ contains a dense countable subgroup $H$. By the countable case there is an
$H$ invariant measure on $X$ satisfying the conditions of the theorem and by
the previous lemma this measure is actually $G$ invariant as needed. ∎

We are now ready to prove theorem 7.2.

###### proof of theorem 7.2.

Consider the measure $d\nu=e^{2h(\xi,\eta)}d\mu^{2}$ on $\partial^{2}G$. Since
we are only considering distinct pairs in $\partial G$ this measure is a Radon
measure (it would not be a Radon measure on $\partial G\times\partial G$). The
theorem will now follow from lemma7.4 if we show $\frac{dg_{*}\nu}{d\nu}\asymp
1$ independently of $g$. Indeed:

$\displaystyle ln\left(\frac{dg_{*}\nu}{d\nu}(\xi,\eta)\right)=$
$\displaystyle
ln\left(\frac{dg_{*}\nu}{dg_{*}\mu^{2}}(\xi,\eta)\frac{dg_{*}\mu^{2}}{d\mu^{2}}(\xi,\eta)\frac{d\mu^{2}}{d\nu}(\xi,\eta)\right)$
$\displaystyle\approx$ $\displaystyle
2h((g^{-1}\xi,g^{-1}\eta)+(g,\xi)+(g,\eta)-|g|-(\xi,\eta))\approx 0$

∎

Recall that given a metric space $(X,d)$ the cross ratio of four distinct
points $x,y,z,w$ is:

$[x,y;z,w]=\frac{d(x,z)d(y,w)}{d(x,w)d(y,z)}$

We will show in the following chapters (theorem 10.5) that the action of $G$
on $(\partial^{2}G,\mu^{2})$ is ergodic. The proof is independent rest of this
chapter so we can use theorem 10.5 and not run in to any circular logic. An
ergodic measure class can not contain more then one invariant measure up to
scalars. Therefore, denoting $F^{2}(\xi,\eta)=(F(\xi),F(\eta))$, we conclude
that $F^{2}_{*}m_{1}=\beta m_{2}$ for some constant $\beta>0$. As a result:

$\frac{dF^{2}_{*}\mu_{1}^{2}}{d\mu_{2}^{2}}(\xi,\eta)\asymp\frac{dF^{2}_{*}\mu_{1}^{2}}{dF^{2}_{*}m_{1}}(\xi,\eta)\frac{dm_{2}}{d\mu_{2}^{2}}(\xi,\eta)\asymp
d_{\epsilon_{1}}(F^{-1}(\xi),F^{-1}(\eta))^{2D_{1}}d_{\epsilon_{2}}(\xi,\eta)^{-2D_{2}}$

One can now calculate the cross ratio in $(\partial G,d_{\epsilon_{1}})$
directly and see that all the Radon-Nikodym derivatives cancel out giving:

$[F(\xi_{1}),F(\xi_{2});F(\eta_{1}),F(\eta_{2})]_{2}\asymp[\xi_{1},\xi_{2};\eta_{1},\eta_{2}]_{1}^{\frac{D_{1}}{D_{2}}}$

which holds on a subset $E$ of full measure in $\partial G^{4}$.

######  Lemma 7.5.

There exists a full measure subset of $E^{\prime\prime}\subseteq\partial G$
such that for $\xi,\eta\in E^{\prime\prime}$:

$d_{\epsilon_{2}}(F(\xi),F(\eta))\prec
d_{\epsilon_{1}}(\xi,\eta)^{\frac{D_{1}}{D_{2}}}$

###### Proof.

Let $(\xi_{2},\eta_{2})$ be such that the fiber
$\\{(\xi_{1},\eta_{1})|(\xi_{1},\xi_{2},\eta_{1},\eta_{2})\in E\\}$ has full
measure in $\partial G\times\partial G$ and let $\rho>0$. Thus the estimate on
the cross ratios implies that for a full measure subset of
$(\xi_{1},\eta_{1})\in B_{\rho}(\eta_{2})^{c}\times B_{\rho}(\xi_{2})^{c}$:

$d_{\epsilon_{2}}(F(\xi_{1}),F(\eta_{1}))\prec_{\xi_{2},\eta_{2},\rho}d_{\epsilon_{1}}(\xi_{1},\eta_{1})^{\frac{D_{1}}{D_{2}}}$

By Fubini’s theorem $(\xi_{2},\eta_{2})$ can be chosen from a full measure
subset which in particular is dense in $\partial G\times\partial G$. By
compactness, $\partial G\times\partial G$ can be covered by finitely many sets
of the form $B_{\rho}(\eta_{2})^{c}\times B_{\rho}(\xi_{2})^{c}$ so that the
above estimate actually holds uniformly over a full measure subset of
$\partial G\times\partial G$. Since $F$ is an almost everywhere defined
isomorphism there is a full measure subset $E^{\prime}$ of $\partial
G\times\partial G$ such that $d_{\epsilon_{2}}(F(\xi),F(\eta))\prec
d_{\epsilon_{1}}(\xi,\eta)^{\frac{D_{1}}{D_{2}}}$.

Let $E^{\prime\prime}$ be the full measure subset of $\partial G$ such that
the fibers over points in $E^{\prime}$ have full measure. If $\xi,\eta\in
E^{\prime\prime}$ and $\zeta$ is in the fiber over both $\xi$ and $\eta$ then:

$d_{\epsilon_{2}}(F(\xi),F(\eta))\leq
d_{\epsilon_{2}}(F(\xi),F(\zeta))+d_{\epsilon_{2}}(F(\zeta),F(\eta))\prec
d_{\epsilon_{1}}(\xi,\zeta)^{\frac{D_{1}}{D_{2}}}+d_{\epsilon_{1}}(\zeta,\eta)^{\frac{D_{1}}{D_{2}}}$

Since $\zeta$ can be taken from a full measure subset we can let it converge
to $\eta$ and we get $d_{\epsilon_{2}}(F(\xi),F(\eta))\prec
d_{\epsilon_{1}}(\xi,\eta)^{\frac{D_{1}}{D_{2}}}$ for all $\xi,\eta\in
E^{\prime\prime}$ as needed. ∎

As a result we see that $F$ is uniformly continuous on $E^{\prime\prime}$ and
thus agrees almost everywhere with a uniformly continuous function $H:\partial
G\to\partial G$. Since $F$ is almost everywhere $G$-equivariant we conclude
that $H$ is $G$-equivariant everywhere so by lemma 2.16 $H$ is trivial.
Finally using the symmetry between $d_{1}$ and $d_{2}$, for all
$\xi,\eta\in\partial G\times\partial G$:

$d_{\epsilon_{2}}(\xi,\eta)\asymp
d_{\epsilon_{1}}(\xi,\eta)^{\frac{D_{1}}{D_{2}}}$

Now we can prove the main theorem of this section:

######  Theorem 7.6.

Let $G$ be a second countable non elementary unimodular locally compact
hyperbolic group and let $d_{1},d_{2}\in\mathcal{D}(G)$. The following are
equivalent:

  1. (1)

The metrics $d_{1}$ and $d_{2}$ are roughly similar.

  2. (2)

The Patterson-Sullivan measures $\mu_{1}$ and $\mu_{2}$ are in the same
measure class.

  3. (3)

The boundary representations $\pi_{1}$ and $\pi_{2}$ are unitarily equivalent.

###### Proof.

If $d_{2}\approx Ld_{1}$ are roughly similar then for a given small enough
visual parameters $\epsilon$ (chosen for simplicity to be the same for both
metrics), the visual metrics $d^{1}_{L\epsilon}$ and $d^{2}_{\epsilon}$ are
bi-Lipschitz and thus produce Hausdorff measures in the same measure class so
by lemma 3.8 $\mu_{1}$ and $\mu_{2}$ are in the same measure class. As
explained in the preliminaries this implies that $\pi_{1}$ and $\pi_{2}$ are
equivalent.

For the implication $(3)\implies(1)$, since $d_{\epsilon_{2}}(\xi,\eta)\asymp
d_{\epsilon_{1}}(\xi,\eta)^{\frac{D_{1}}{D_{2}}}$ the corresponding $D_{i}$
-dimensional Hausdorff measures are in the same measure class with Radon-
Nikodym derivatives bounded away from $0$ and $\infty$. By lemma 3.8 the same
is true for $\mu_{1}$ and $\mu_{2}$. Therefore for any $g\in G$:

$e^{h_{1}(2(g,\xi)_{1}-|g|_{1})}\asymp e^{h_{2}(2(g,\xi)_{2}-|g|_{2})}$

By lemma 2.13, after taking suprema over $\xi$ we get that
$e^{h_{1}|g|_{1}}\asymp e^{h_{2}|g|_{2}}$ so:

$|g|_{1}\approx\frac{h_{2}}{h_{1}}|g|_{2}$

as needed. ∎

We have now finished the proof of theorem 1.2, assuming theorem 10.5 holds.

##  8\. Type I Hyperbolic Groups

One of the original motivations for generalizing Garncarek’s work [15] to the
locally compact setting was that such a generalization would strengthen the
results of Caprace, Klantar and Monod in [11]. This is pointed out explicitly
in [11, Remark 5.6]. We describe the strengthened results here.

######  Definition 8.1.

A locally compact group $G$ is of type I if any two weakly equivalent (see [4,
Appendix F.1]) irreducible unitary representations are unitarily equivalent.

If $G$ is a second countable totally disconnected unimodular hyperbolic
locally compact group then it follows from theorem 1.1 that for any compact
open subgroup $U$ and any Cayley-Abels graph on $G/U$ the Patterson-Sullivan
representation corresponding to the induced (pseudo)metric on $G$ is
irreducible. Similarly by theorem 1.2 the identity on $G/U$ is a rough
similarity between two Cayley-Abels graphs if and only if the corresponding
Patterson-Sullivan representations are unitarily equivalent. These are exactly
conditions (G1) and (G2) mentioned in [11, Remark 5.6], thus it follows from
[11, Remark 5.6] that [11, Theorem B],[11, Theorem K] hold in greater
generality. Explicitly:

######  Theorem 8.2.

Let $G$ be a second countable unimodular hyperbolic locally compact group. If
$G$ is of type I then $G$ admits a cocompact amenable subgroup.

######  Theorem 8.3.

Let $G$ be a non-amenable second countable unimodular hyperbolic locally
compact group. Let $\mu$ be a quasi invariant measure on $\partial G$, $\pi$
the corresponding Koopman representation and $C^{*}_{\pi}(G)$ the image of the
group $C^{*}$-algebra $C^{*}(G)$ under $\pi$. The following are equivalent:

  * •

$C^{*}_{\pi}(G)$ contains a non-zero CCR closed two sided ideal.

  * •

$C^{*}_{\pi}(G)$ is GCR.

  * •

$C^{*}_{\pi}(G)$ is CCR.

  * •

$C^{*}_{\pi}(G)$ consists entirely of compact operators.

  * •

G has a cocompact amenable subgroup.

Note that the last item of the proposition is independent of $\mu$.

It is an interesting question whether the unimodularity assumption in these
theorems can be replaced by the weaker assumption of non-amenability. This
would follow as above if one would show the conclusions of theorems 1.1 and
1.2 hold for non amenable hyperbolic groups.

##  9\. The Geodesic Flow

We retain with the assumptions from the last sections that $G$ is second
countable locally compact hyperbolic and unimodular. In the special case when
$G$ is a discrete group acting freely properly and co-compactly on a
negatively curved simply connected manifold $M$ one can associate to this
action the geodesic flow on the unit tangent bundle of the quotient
$T^{1}N=G\backslash T^{1}M$. In this restricted setting $T^{1}M$ is
canonically identified with the space of pointed geodesics in $M$. By looking
at the endpoints of each geodesic we see that this space is a line bundle over
$\partial^{2}G$. Measure theoretically we can identify this bundle with
$\partial^{2}G\times\mathbb{R}$ and the action of $G$ on on this space is
given by a certain cocycle $\tau:G\times\partial^{2}G\to\mathbb{R}$. This
point of view can be generalized to our setting and will allow us to define a
geodesic flow for general $(G,d)$. In the case where $G$ is a discrete group
this is done in [15, Appendix A.1] and [2, Section 3]. We will encounter some
added measure theoretic difficulties. The two major complications are the
following:

  * •

If $G$ is uncountable not every null set is contained in a $G$-invariant null
set so one must take care to ignore only invariant null sets.

  * •

If $G$ is uncountable a fundamental domain for a non-singular action of $G$
will typically be a null set (which is of course not $G$-invariant).

Define $\sigma,\rho:G\times\partial G\to\mathbb{R}$ and
$\tau:G\times\partial^{2}G\to\mathbb{R}$ by:

$\sigma(g,\xi)=2(g^{-1},\xi)-|g|$
$\rho(g,\xi)=\frac{1}{h}ln\frac{dg^{-1}_{*}\mu}{d\mu}(\xi)$
$\tau(g,\xi,\eta)=\frac{\rho(g,\eta)-\rho(g,\xi)}{2}\approx(g^{-1},\eta)-(g^{-1},\xi)$

By the chain rule for Radon-Nikodym derivatives $\rho$ and $\tau$ are
cocycles, i.e. they satisfy the cocycle equation:

$c(gh,x)=c(g,hx)+c(h,x)$

By [22, Appendix B.9] we can assume that $\rho$ and $\tau$ are strict
cocycles, i.e. that the cocycle equation is satisfied everywhere (as opposed
to almost everywhere). $\sigma$ is a strict almost cocycle in the sense that
for all every $g,h$ and $\xi$:

$\sigma(gh,\xi)\approx\sigma(g,h\xi)+\sigma(h,\xi)$

uniformly in $g,h,\xi$.

Finally, by quasi-conformality of $\mu$ it follows that for every $g\in G$,
for almost every $\xi\in\partial G$:

$\rho(g,\xi)\approx\sigma(g,\xi)$ (3)

By appendix A.1, after replacing $\rho$ by a strictly cohomologous cocycle we
can assume that estimate 3 holds on a $G$-invariant full measure subset
$A\subseteq\partial G$. Abusing notation we call will denote this new cocycle
again by $\rho$. We now get also that for all $g\in G$ and $(\xi,\eta)$ in a
full measure $G$-invariant set $S$ (for example $S=A^{2}-\Delta A$) we have:

$\tau(g,\xi,\eta)\approx\frac{\sigma(g,\eta)-\sigma(g,\xi)}{2}$

We will need the following lemma in the next section:

######  Lemma 9.1.

For every $M>0$ there exists a constant $C_{M}$ such that if $(\xi,\eta)\in S$
such that $(\xi,\eta),(g\xi,g\eta)<M$ then
$|\tau(g,\xi,\eta)-\sigma(g,\eta)|<C_{M}$.

###### Proof.

Since $\sigma(g,\eta)\approx\rho(g,\eta)$ we have that
$\tau(g,\xi,\eta)-\sigma(g,\eta)\approx-\frac{\rho(g,\eta)+\rho(g,\xi)}{2}$.
Using the chain rule for Radon Nikodym derivatives:

$1=\frac{dg^{-1}_{*}m}{dm}(\xi,\eta)=\frac{dm}{d\mu^{2}}(g\xi,g\eta)\frac{dg^{-1}_{*}\mu^{2}}{d\mu^{2}}(\xi,\eta)\frac{d\mu^{2}}{dm}(\xi,\eta)$

Taking the logarithm of both sides and recalling that
$\frac{dm}{d\mu^{2}}(\xi,\eta)\asymp e^{2h(\xi,\eta)}$ we see that
$|\frac{\rho(g,\eta)+\rho(g,\xi)}{2}|\approx|(\xi,\eta)-(g\xi,g\eta)|\leq 2M$
as needed. ∎

Fix an ergodic p.m.p action of $G$ on a standard probability space
$(\Omega,\omega)$, we introduce this space to increase the generality of the
next section but it will have very little effect on the discussion and for our
purposes one can take $\Omega$ to be trivial. Recall the invariant measure $m$
constructed on $\partial^{2}G$. Denote the Lebesgue measure on $\mathbb{R}$ by
$\ell$. Using the cocycle $\tau$ we can define an (infinite) measure
preserving action of $G$ on the space
$(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$ by:

$g(\xi,\eta,t,w)=(g\xi,g\eta,t+\tau(g,\xi,\eta),gw)$

This action commutes with the $\mathbb{R}$-flow defined by:

$\Phi^{s}(\xi,\eta,t)=(\xi,\eta,t+s)$

Denote:

$D_{\theta,k}=\\{(\xi,\eta,t,w)\in
S\times(-k,k)\times\Omega|d_{\epsilon}(\xi,\eta)>\theta\\}$

We call sets contained in some $D_{\theta,k}$ bounded. Being bounded is
equivalent to having precompact projection in $\partial^{2}G\times\mathbb{R}$.
Bonded sets have finite $m\times\ell\times\omega$ measure.

Similarly to the proofs in of [15, Appendix A.2,A.3] one proves the following
two lemmas:

######  Lemma 9.2.

For any $\theta,k>0$, $\\{g\in G|gD_{\theta,k}\cap D_{\theta,k}\neq\phi\\}$ is
bounded in the metric $d$ and precompact in $G$.

###### Proof.

Suppose that $gD_{\theta,k}\cap D_{\theta,k}\neq\phi$. There exists
$(\xi,\eta,t,w)\in D_{\theta,k}$ such that
$(g\xi,g\eta,t+\tau(g,\xi,\eta),gw)\in D_{\theta,k}$. Therefore,
$(\xi,\eta),(g\xi,g\eta)\approx 0$ and
$|(g^{-1},\eta)-(g^{-1},\xi)|\approx|\tau(g,\xi,\eta)|\approx 0$. Now
$(\xi,\eta)\gtrsim\min\\{(\xi,g^{-1}),(g^{-1},\eta)\\}$ but $(\xi,\eta)\approx
0$ and $(g^{-1},\eta)\approx(g^{-1},\xi)$ so
$(\xi,g^{-1})\approx(g^{-1},\eta)\approx 0$. Using lemma 2.12 we see that:

$(g\xi,g\eta)\gtrsim\min\\{(g\xi,g),(g,\eta)\\}\approx|g|-\max\\{(g^{-1},\xi),((g^{-1},\eta)\\}\approx|g|$

But $(g\xi,g\eta)\approx 0$ so $|g|\approx 0$ as needed. ∎

######  Lemma 9.3.

For large enough $\theta$ and $k$, $D_{\theta,k}$ intersects every $G$ orbit
in $S\times\mathbb{R}\times\Omega$.

###### Proof.

Let $(\xi,\eta,t,w)\in S\times\mathbb{R}\times\Omega$ and let
$\gamma:(-\infty,\infty)\to G$ be a rough geodesic with
$\gamma(\infty)=\eta,\gamma(-\infty)=\xi$. For any $r\in\mathbb{R}$ we have
that $(\xi,\eta)_{\gamma(r)}\approx 0$ and thus
$(\gamma(r)^{-1}\xi,\gamma(t)^{-1}\eta)\approx 0$. In addition:

$\displaystyle\sigma(\gamma(r)^{-1},\eta)$ $\displaystyle=$ $\displaystyle
2(\gamma(r),\eta)-|\gamma(r)|\approx\liminf_{s\to\infty}|\gamma(r)|-(s-r)$
$\displaystyle=$ $\displaystyle
r+\liminf_{s\to\infty}(|\gamma(s)|-s)=r+\sigma(\gamma(0)^{-1},\eta)$

and similarly $\sigma(\gamma(r)^{-1},\xi)\approx-r+\sigma(\gamma(0)^{-1},\xi)$
so:

$\tau(\gamma(r)^{-1},\xi,\eta)\approx r+\tau(\gamma(0)^{-1},\xi,\eta)$

Putting everything together we see that for
$r=-\tau(\gamma(0)^{-1},\xi,\eta)-t$:
$(\gamma(r)^{-1}\xi,\gamma(r)^{-1}\eta)\approx 0$ and
$\tau(\gamma(r)^{-1},\xi,\eta)\approx 0$. Therefore
$\gamma(r)^{-1}(\xi,\eta,t,w)$ is in some $D_{\theta,k}$ with $\theta$ and $k$
independent of $(\xi,\eta,t,w)$. ∎

######  Lemma 9.4.

If $U\subset G$ is bounded then for any $\theta$ and $k$, $UD_{\theta,k}$ is
bounded.

###### Proof.

Given $g\in G$ and $(\xi,\eta)\in\partial^{2}G$ we have that
$|\tau(g,\xi,\eta)|\approx|(g^{-1},\xi)-(g^{-1},\eta)|\lesssim 2|g|$ and
$|(g\xi,g\eta)-(\xi,\eta)|\approx|(\xi,\eta)_{g^{-1}}-(\xi,\eta)|\lesssim|g|$.
Together with the boundedness of $U$ this shows that for any $\theta$ and $k$
there exist $\theta^{\prime}$ and $k^{\prime}$ such that
$UD_{\theta,k}\subseteq D_{\theta^{\prime},k^{\prime}}$. ∎

Intuitively one should think of lemma 9.2 and lemma 9.3 as saying that the $G$
action on $S\times\mathbb{R}\times\Omega$ is proper and co-compact, although
formally this is a measurable action and not a continuous topological action.
In the same spirit we have the following theorem:

######  Theorem 9.5.

The $G$ action on $S\times\mathbb{R}\times\Omega$ admits a bounded Borel cross
section, i.e. a bounded Borel subset $\hat{X}$ intersecting each $G$ orbit
exactly once.

###### Proof.

By [10, Proposition 5.10] $G$ has a maximal normal compact subgroup $W$ and
$G/W$ is either a virtually connected rank one simple adjoint Lie group or
$G/W$ is totally disconnected.

In case $G/W$ is connected the action of $G$ on $\partial G$ is transitive so
there is a unique invariant measure class on $\partial G$ and $\mu$ is in this
measure class so the cocycle $\tau$ is independent of $d$ up to cohomology and
the actions on $S\times\mathbb{R}\times\Omega$ given by different $d$ are
conjugate. For the specific (psudo)metric pulled back from the corresponding
symmetric space this action is simply the action on the unit tangent bundle of
the symmetric space which is known to be transitive so any point is a cross
section.

In case $G/W$ is totally disconnected, by van Dantzig’s theorem $G$ admits a
compact open subgroup $U$. Since $U$ is compact it is bounded in the metric
$d$.

By [22, Corrolary 2.1.21] and [22, Appendix A.7] the $U$ action on
$S\times\mathbb{R}\times\Omega$ admits a Borel cross section $\Delta$. By
lemma 9.3 there exists some $D_{\theta^{\prime},k^{\prime}}$ intersecting
every $G$ orbit and by lemma 9.4 there exist $\theta,k$ such that
$UD_{\theta^{\prime},k^{\prime}}\subseteq D_{\theta,k}$. Since every point has
a unique point of $\Delta$ in its $U$ orbit every $G$ orbit intersects
$D_{\theta,k}\cap\Delta$ non trivially. By lemma 9.2 there exists a compact
set $F\subset G$ such that if $g\notin F$, $gD_{\theta,k}\cap
D_{\theta,k}=\phi$. We will importantly use the fact that this intersection is
empty and not only null, this is the reason we are restricting ourselves from
$\partial^{2}G$ to $S$. Enlarging $F$ we can assume without loss of generality
that $F=\bigcup_{i}Ug_{i}$ is a finite union of right cosets of U. If $x\in
S\times\mathbb{R}\times\Omega$ and $g\in G$ such that $x,gx\in
D_{\theta,k}\cap\Delta$ then $g\in Ug_{i}$ for some $i$, but
$|Ug_{i}x\cap\Delta|=1$. Therefore the intersection of each $G$ orbit with
$D_{\theta,k}\cap\Delta$ is finite and non empty.

To sum up, every $G$ orbit intersects $D_{\theta,k}\cap\Delta$ and the
restriction of the $G$ orbit equivalence relation to $D_{\theta,k}\cap\Delta$
has finite equivalence classes. Using the finite Borel selection theorem we
deduce that there is a Borel subset $\hat{X}\subseteq D_{\theta,k}\cap\Delta$
intersecting every $G$ orbit exactly once as needed. This set is bounded since
it is contained in $D_{\theta,k}$. ∎

We now get a Borel isomorphism between the the space
$X=(S\times\mathbb{R}\times\Omega)//G$ of $G$-ergodic components and $\hat{X}$
by restricting the projection $p:S\times\mathbb{R}\times\Omega\to X$ to
$\hat{X}$. This identifies the orbit space of $G$ with the space of ergodic
components. Since the flow $\Phi$ on $(S\times\mathbb{R}\times\Omega)$
commutes with $G$ it descends to a flow $\phi$ on $X$ such that
$p\circ\Phi^{s}=\phi^{s}\circ p$.

Since $G$ is unimodular and for any compact $U\subseteq G$, $U\hat{X}$ is
bounded, by theorems B.1 and B.2 there exists a unique measure $\nu$ on $X$ in
the measure class of $p_{*}(m\times\ell\times\omega)$ which is invariant under
the flow $\phi$ and such that for any $f\in
L^{1}(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$:

$\int f(z)dm\times\ell\times\omega(z)=\int\int f(gz)d\lambda(g)d\nu(p(z))$ (4)

(Note that the inner integral on the right hand side depends only on $p(z)$
and not on $z$ because $G$ is unimodular.)

Since for any compact $U\subset G$, $U\hat{X}$ is bounded, it follows form
equation 4 that $\nu$ is a finite measure and after re-scaling $m$ we can
assume without loss of generality that $\nu$ is a probability measure. If we
take $\Omega$ to be trivial then we call the associated flow $(X,\nu,\phi)$
the geodesic flow of $(G,d)$.

##  10\. Double Ergodicity

We keep all the notation and assumptions from the previous section. In this
section we will prove that the action of $G$ on $\partial^{2}G$ is ergodic and
even weak mixing, meaning that for any ergodic p.m.p action
$G\operatorname{\curvearrowright}(\Omega,\omega)$ the diagonal action on
$(\partial^{2}G\times\Omega,m\times\omega)$ is ergodic. We will use a classic
argument of E. Hopf and adapt the proof in [2, Section 4] to the locally
compact setting. The author would like to thank Uri Bader and Alex Furman for
sending him a forthcoming unpublished version of [2] on which our arguments
are very closely based.

The space $(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$ is
equipped with the commuting measure preserving actions of $G$ and the
$\mathbb{R}$-flow $\Phi$ discussed in the previous section. The space of
$\mathbb{R}$-ergodic components is $(S\times\Omega,[m\times\omega])$ and in
its canonical measure class it supports the $G$-invariant infinite measure
$m\times\omega$. The space $(X,[\nu])$ of $G$-ergodic components has the
natural $\mathbb{R}$-flow $\phi$ and contains the finite measure $\nu$ in its
canonical measure class. $\phi$ is $\mathbb{R}$-invariant and satisfies
equation 4. After re-scaling $m$ we can further assume that $\nu$ is a
probability measure. The $\mathbb{R}$-ergodic components of $(X,\nu)$ , the
$G$-ergodic components of $(S\times\Omega,m\times\omega)$ and the
$G\times\mathbb{R}$-ergodic components of
$(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$ are all given by the
same space $(Y,[\beta])$. In the measure class $[\beta]$ we choose the measure
$\beta$ which is the image of $\nu$. We thus obtain the following commutative
diagram:

${{(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)}}$${{(X,\nu)}}$${{(S\times\Omega,m\times\omega)}}$${{(Y,\beta)}}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{u}$$\scriptstyle{\bar{p}}$$\scriptstyle{v}$
(5)

The maps $p$ and $v$ are the $G$-ergodic components maps, the maps $q$ and $u$
are the $\mathbb{R}$-ergodic components maps and the map $u\circ p=v\circ q$
is the $G\times\mathbb{R}$-ergodic components map. The maps $p$ and $q$ are
$\mathbb{R}$-equivariant and $G$-equivariant respectively. Ergodicity of $G$
on $(S\times\Omega,m\times\omega)$, ergodicity of $\mathbb{R}$ on $(X,\mu)$
and ergodicity of $G\times\mathbb{R}$ on
$(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$ are all equivalent
to $\beta$ being supported on a single point.

In the previous section we constructed a bounded cross section $\hat{X}$ for
the $G$ action on $(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$.
The map $\bar{p}:X\to S\times\mathbb{R}\times\Omega$ describes the
corresponding section of $p$. Unlike the case of discrete $G$ the measure
$\bar{p}_{*}\nu$ might be singular to $m\times\ell\times\omega$. Given $x\in
X$ we denote

$\bar{p}(x)=(x_{-},x_{+},t_{x},\omega_{x})$

The maps $p,q,u$ and $v$ can be used to push forward finite signed measure
which are absolutely continuous with respect to the given measure classes. By
the Radon-Nikodym theorem the space of finite signed measures absolutely
continuous with respect to a given measure is identified with the space of
integrable functions. Under these isomorphisms we obtain the operators
$P,Q,U,V$ in the following commutative diagram:

${{L^{1}(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)}}$${{L^{1}(X,\nu)}}$${{L^{1}(S\times\Omega,m\times\omega)}}$${{L^{1}(Y,\beta)}}$$\scriptstyle{P}$$\scriptstyle{Q}$$\scriptstyle{U}$$\scriptstyle{V}$$\scriptstyle{R_{\theta}}$
(6)

We will now give explicit descriptions of $Q,P$ and $U$. The operator $Q$ is
given by integration over the $\mathbb{R}$ coordinate, For any $f\in
L^{1}(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$:

$Q(f)(\xi,\eta,w)=\int f(\xi,\eta,t,w)dt$

By equation 4, $P$ is given by integration over $G$ orbits, or equivalently
over fibers of $p$. Given $f\in
L^{1}(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$:

$P(f)(p(\xi,\eta,t,w))=\int f(g(\xi,\eta,t,w))d\lambda(g)$

Note that since $G$ is unimodular the right hand side indeed depends only on
$p(\xi,\eta,t,w)$ (this fact is critical in the construction of the measure
$\nu$).

Using Birkhoff’s ergodic theorem we see that $U$ is given by averaging over
$\mathbb{R}$ orbits in $X$. For $f\in L^{1}(X,\nu)$, for almost every $x\in
X$, for any $a\leq b\in\mathbb{R}$:

$U(f)(u(x))=\frac{1}{T}\int_{a}^{b+T}f(\phi^{-t}x)dt$

Our next goal will be to obtain an explicit description of $V$. For this we
need to explain the doted arrow $\mathbb{R}_{\theta}$ in diagram 6. If
$\theta\in L^{1}(\mathbb{R})$ is a positive kernel, i.e
$\theta:\mathbb{R}\to[0,\infty)$ and:

$\int\theta(t)dt=1$

We define $R_{\theta}:L^{1}(S\times\Omega,m\times\omega)\to
L^{1}(S\times\mathbb{R}\times\Omega,m\times\ell\times\omega)$ by:

$\mathbb{R}_{\theta}(f)(\xi,\eta,t,w)=\theta(t)f(\xi,\eta,w)$

Clearly $Q\circ R_{\theta}=Id$. Since $V\circ Q=U\circ P$ we can precompose
with $R_{\theta}$ to get that $V=U\circ P\circ R_{\theta}$. Given $f\in
L^{1}(S\times\Omega,m\times\omega)$ denote:

$\bar{f}_{\theta}(x)=P\circ R_{\theta}(f)=\int
R_{\theta}f(g\bar{p}(x))d\lambda(g)=\int\theta(t_{x}+\tau(g,x_{-},x_{+}))f(gx_{-},gx_{+},gw_{x})d\lambda(g)$
(7)

We also denote:

$\bar{f}_{[0,1]}=\bar{f}_{\mathbbm{1}_{[0,1]}}$

Using $R_{\theta}$ we obtain the following description of $V$. For any $a\leq
b$, for almost every $y\in Y$, if $y=u(x)$:

$V(f)(y)=\frac{1}{T}\int_{a}^{b+T}\bar{f}_{[0,1]}(\phi^{-t}x)dt$ (8)

While this formula is explicit it describes the function $V(f)$ in terms of
the parameter $x\in X$. We would like to obtain a description of $V$ in terms
of a points $(\xi,\eta,w)\in S\times\Omega$. For this purpose we introduce the
following averaging operators. For any $a\leq b$ and $f\in
L^{1}(S\times\Omega,m\times\omega)$ define
$I_{a}^{b}(f),J_{a}^{b}(f):S\times\Omega\to[0,\infty]$ by:

$I_{a}^{b}(f)(\xi,\eta,w)=\frac{1}{b-a}\int_{\\{g\in
G|\tau(g,\xi,\eta)\in[a,b]\\}}f(g\xi,g\eta,gw)d\lambda(g)$

$J_{a}^{b}(f)(\xi,\eta,w)=\frac{1}{b-a}\int_{\\{g\in
G|\sigma(g,\eta)\in[a,b]\\}}f(g\xi,g\eta,gw)d\lambda(g)$

We prove the following ergodic theorem describing $V$:

######  Theorem 10.1.

For any $f\in L^{1}(S\times\Omega,m\times\omega)$, for almost every
$(\xi,\eta,w)$ the values $I_{a}^{b}(f)(\xi,\eta,w)$ and
$J_{a}^{b}(f)(\xi,\eta,w)$ are finite for any interval $[a,b]$ and if
$y=v(\xi,\eta,w)$ then:

$V(f)(y)=\lim_{T\to\infty}I_{a}^{b+T}(f)(\xi,\eta,w)$

If in addition $f$ is supported on a bounded set then:

$V(f)(y)=\lim_{T\to\infty}J_{a}^{b+T}(f)(\xi,\eta,w)$

The proof will require two lemmas.

######  Lemma 10.2.

For any two positive kernels $\theta_{1},\theta_{2}$ and $f\in
L^{1}(S\times\Omega,m\times\omega)$:

$\bar{f}_{\theta_{2}*\theta_{1}}=\int_{-\infty}^{\infty}\theta_{2}(s)\bar{f}_{\theta_{1}}\circ\phi^{-s}ds$

###### Proof.

Notice that:

$R_{\theta_{2}*\theta_{1}}f(\xi,\eta,t,w)=\int_{-\infty}^{\infty}\theta_{2}(s)\theta_{1}(t-s)f(\xi,\eta,w)ds=\int_{-\infty}^{\infty}\theta_{2}(s)R_{\theta_{1}}f\circ\Phi^{-s}(\xi,\eta,t,w)ds$

We finish by applying $P$ to both sides. ∎

######  Lemma 10.3.

There exists a constant $c\geq 0$ such that for every $m\times\omega$
integrable function $f:S\times\Omega\to[0,\infty)$, for every interval $[a,b]$
with $a+c+1\leq b$ and for every $x\in X$:

$\frac{1}{b-a}\int_{a+c}^{b-c-1}\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt\leq
I_{a}^{b}(f)(x_{-},x_{+},w_{x})\leq\frac{1}{b-a}\int_{a-c-1}^{b+c}\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt$

###### Proof.

Denote $\theta_{a}^{b}=\mathbbm{1}_{[a,b]}*\mathbbm{1}_{[0,1]}$ and notice
that if $a+1\leq b$ then $\theta_{a}^{b}$ interpolates linearly between the
values $0$ on $(-\infty,a]\cup[b+1,\infty)$ and $1$ on $[a+1,b]$. In
particular $\mathbbm{1}_{[a+1,b]}\leq\theta_{a}^{b}\leq\mathbbm{1}_{[a,b+1]}$.
Recall that the cross section $\hat{X}$ is bounded, therefore there exists a
constant $c$ such that for any $x\in X$, if
$\bar{p}(x)=(x_{-},x_{+},t_{x},w_{x})$ then $|t_{x}|<c$. Let $x\in X$ and
$a,b\in\mathbb{R}$ such that $a+c+1\leq b$, using lemma 10.2 and equation 7 we
see that:

$\displaystyle\int_{a}^{b}\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\mathbbm{1}_{[a,b]}(t)\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt=\bar{f}_{\theta_{a}^{b}}(x)$
$\displaystyle=$
$\displaystyle\int_{G}\theta_{a}^{b}(t_{x}+\tau(g,x_{-},x_{+}))f(gx_{-},gx_{+},gw_{x})d\lambda(g)$

Since $|t_{x}|<c$ and
$\mathbbm{1}_{[a+1,b]}\leq\theta_{a}^{b}\leq\mathbbm{1}_{[a,b+1]}$ we conclude
that:

$\frac{b-a-2c-1}{b-a}I_{a+c+1}^{b-c}f(x_{-},x_{+},w_{x})\leq\frac{1}{b-a}\int_{a}^{b}\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt\leq\frac{b-a+2c+1}{b-a}I_{a-c}^{b+c+1}f(x_{-},x_{+},w_{x})$

Rewriting these inequalities we get the desired result. ∎

###### proof of theorem 10.1.

We first prove the claim for the operator $I$. Let
$f:S\times\Omega\to[0,\infty)$ be $m\times\omega$ integrable. By Birkhoff’s
ergodic theorem there exists a full measure subset $X_{0}\subseteq X$ such
that for $x\in X_{0}$ and any $a<b$:

$V(f)(u(x))=\lim_{T\to\infty}\frac{1}{T}\int_{a}^{b+T}\bar{f}_{[0,1]}(\phi^{-t}x)dt$

$X_{0}$ is an $\mathbb{R}$-invariant subset so $p^{-1}(X_{0})$ is a
$G\times\mathbb{R}$-invariant subset in $S\times\mathbb{R}\times\Omega$ and
$A_{f}=q(p^{-1}(X_{0}))$ is a $G$-invariant Borel subset of $S\times\Omega$.
We will show that the desired convergence holds on $A_{f}$.

Let $(\xi,\eta,w)\in A_{f}$, consider $x=p(\xi,\eta,0,w)\in X_{0}$. There
exists $h\in G$ such that
$(\xi,\eta,0,w)=h(x_{-},x_{+},t_{x},w_{x})=h\bar{p}(x)$. Denoting
$s=\tau(h,x_{-},x_{+})$ we have that $\tau(g,\xi,\eta)=\tau(gh,x_{-},x_{+})-s$
so $\tau(g,\xi,\eta)\in[a,b]$ if and only if
$\tau(gh,x_{-},x_{+})\in[a+s,b+s]$. Therefore:

$\displaystyle I_{a}^{b}(f)(\xi,\eta,w)=\frac{1}{b-a}\int_{\\{g\in
G|\tau(g,\xi,\eta)\in[a,b]\\}}f(ghx_{-},ghx_{+},ghw_{x})d\lambda(g)$
$\displaystyle=$ $\displaystyle\frac{1}{b-a}\int_{\\{g\in
G|\tau(gh,x_{-},x_{+})\in[a+s,b+s]\\}}f(ghx_{-},ghx_{+},ghw_{x})d\lambda(g)=I_{a+s}^{b+s}(f)(x_{-},x_{+},w_{x})$

Note that the last equality uses the assumption that $G$ is unimodular. By
lemma 10.3, for any $a,b$ such that $a+c+1<b$:

$\frac{1}{b-a}\int_{a+s+c}^{b+s-c-1}\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt\leq
I_{a+s}^{b+s}(f)(x_{-},x_{+},w_{x})\leq\frac{1}{b-a}\int_{a+s-c-1}^{b+s+c}\bar{f}_{[0,1]}\circ\phi^{-t}(x)dt$

Taking $b$ to $\infty$ we see that as needed:

$\displaystyle V(f)(u(x))$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{a}^{b+T}\bar{f}_{[0,1]}(\phi^{-t}x)dt$
$\displaystyle=$
$\displaystyle\lim_{T\to\infty}I_{a+s}^{b+T+s}(f)(x_{-},x_{+},w_{x})=\lim_{T\to\infty}I_{a}^{b+T}(f)(\xi,\eta,w)$

Since $f$ is non negative this also implies the finiteness of
$I_{a}^{b}(f)(\xi,\eta,w)$ for any interval $[a,b]$ and $(\xi,\eta,w)\in
A_{f}$. The case of general $f$ follows by linearity.

We now turn to the proof for the operator $J$. Denote $K_{n}=\\{(\xi,\eta)\in
S|(\xi,\eta)>n\\}$. Let $f:S\to[0,\infty)$ be a $m\times\omega$ integrable
function supported on bounded set. For large enough $n$ the support of $f$ is
contained in $K_{n}$. Recall that by lemma 9.1 there exist constants $C_{n}$
such that if $(\xi,\eta),(g\xi,g\eta)\in K_{n}$ then:

$|\tau(g,\xi,\eta)-\sigma(g,\eta)|\leq C_{n}$

It follows that for large $n$, any $(\xi,\eta,w)\in K_{n}\times\Omega$ and any
$a<b$:

$\frac{b-a-2C_{n}}{b-a}I_{a+C_{n}}^{b-C_{n}}(f)(\xi,\eta,w)\leq
J_{a}^{b}(f)(\xi,\eta,w)\leq\frac{b-a+2C_{n}}{b-a}I_{a-C_{n}}^{b+C_{n}}(f)(\xi,\eta,w)$

Letting $b$ tend to $\infty$ we get that for every $n$, $(\xi,\eta,w)\in
A_{f}\cap K_{n}\times\Omega$ and $a<b$:

$\lim_{T\to\infty}J_{a}^{b+T}(f)(\xi,\eta,w)=V(f)(v(\xi,\eta,w))$

Since $n$ is arbitrary and $K_{n}$ exhaust the space we see that the
convergence actually holds on all of $A_{f}$. This also implies that
$J_{a}^{b}(f)(\xi,\eta,w)$ is finite for any interval $[a,b]$. Finally the
case of general $f$ follows by linearity. ∎

From this point onward it will be convenient to use the space $\partial^{2}G$
and not $S$. Since $S$ is $G$-invariant and of full measure all the results
proved thus far transfer immediately (although in the proof of the existence
of $\hat{X}$ it was important that certain relations hold everywhere and not
only almost everywhere, and now $\hat{X}$ will intersect only almost every
$G$-orbit).

The last ingredient we are missing for the proof of ergodicity is the
following contraction lemma from [2, Lemma 2.6]:

######  Lemma 10.4.

([2, Lemma 2.6]) For any compact $K\subset\partial^{2}G$ there exists a
constant $C_{K}$ such that for any $\xi,\xi^{\prime},\eta\in\partial G$ and
$g\in G$ satisfying :

$(\xi,\eta),(\xi^{\prime},\eta),(g\xi,g\eta)\in K,\hskip
20.0pt\sigma(g,\eta)>0$

we have:

$\sigma(g,\xi),\sigma(g,\xi^{\prime})\in[-\sigma(g,\eta)-C_{K},-\sigma(g,\eta)+C_{K}],\hskip
20.0ptd_{\epsilon}(g\xi,g\xi^{\prime})<e^{-\epsilon\sigma(g,\eta)+C_{K}}$

###### Proof.

Because $(\xi,\eta),(\xi^{\prime},\eta),(g\xi,g\eta)\in K$ and $K$ is compact
we have that $(\xi,\eta)$, $(\xi^{\prime},\eta)$, $(g\xi,g\eta)\approx_{K}0$.
Since $\sigma(g,\eta)>0$ we see that $(g^{-1},\eta)>\frac{|g|}{2}$ but
$\min\\{(\xi,g^{-1}),(g^{-1},\eta)\\}\lesssim(\xi,\eta)\approx_{K}0$ so we
conclude that $(g^{-1},\xi)\approx_{K}0$. By lemma 2.12 we deduce:

$0\approx_{K}(g\xi,g\eta)\gtrsim\min\\{(g\xi,g),(g,g\eta)\\}\approx|g|-\max\\{(g^{-1},\xi),(g^{-1},\eta)\\}$

Since $(g^{-1},\xi)\approx 0$ it follows that $(g^{-1},\eta)\approx_{K}|g|$.
Now note that
$\min\\{(\xi,g^{-1}),(g^{-1},\eta)\\}\lesssim(\xi,\eta)\approx_{K}0$ so since
$(g^{-1},\eta)\approx_{K}|g|$ it follows that $(\xi,g^{-1})\approx_{K}0$ and
similarly $(\xi^{\prime},g^{-1})\approx_{K}0$. Therefore:

$\sigma(g,\xi)\approx_{K}\sigma(g,\xi^{\prime})\approx_{K}-\sigma(g,\eta)\approx_{K}-|g|$

Using 2.12 again we see that $(g\xi,g)\approx|g|-(g^{-1},\xi)\approx_{K}|g|$
and similarly $(g\xi^{\prime},g)\approx_{K}|g|$ so finally:

$(g\xi,g\xi^{\prime})\gtrsim\min\\{(g\xi,g),(g,g\xi^{\prime})\\}\approx_{K}|g|\approx_{K}\sigma(g,\eta)$

as needed.

∎

Now we will finally prove ergodicity.

######  Theorem 10.5.

For any ergodic p.m.p action $G\operatorname{\curvearrowright}(\Omega,\omega)$
the diagonal action on $(\partial^{2}G\times\Omega,m\times\omega)$ is ergodic.

###### Proof.

To show ergodicity we will show that $L^{1}(Y,\beta)$ is one dimensional. In
order to do this we will show that for any $f\in
L^{1}(\partial^{2}\times\Omega,m\times\omega)$, $V(f)$ is constant. Since $V$
is a bounded operator it suffices to show this for $f$ from a dense subspace
of $L^{1}(\partial^{2}\times\Omega,m\times\omega)$. We will use the image of
the space $C_{c}(\partial^{2}G,m)\otimes L^{1}(\Omega,\omega)$ under the
injection $\phi\otimes\psi\mapsto\phi\cdot\psi$. It is enough to show the
image of pure tensors under $V$ is constant. Let $f=\phi\cdot\psi$ with
$\phi\in C_{c}(\partial^{2}G,m),\psi\in L^{1}(\Omega,\omega)$ and
$\bar{f}=V(f)\circ v$. We will show that $\bar{f}$ is constant. Note that
$\bar{f}$ need not be in $L^{1}(\partial^{2}G\times\Omega,m\times\omega)$. Our
strategy will be to prove that $\bar{f}(\xi,\eta,w)$ is independent of $\xi$
and by symmetry also independent of $\eta$ so $\bar{f}$ is actually given by a
$G$-invariant function of $w$ which is constant by ergodicity.

Fix a compact set $K\subset\partial^{2}G$ with $supp(\phi)\subset int(K)$ and
$\delta>0$. Denote $h=\mathbbm{1}_{K}\cdot|\psi|\in
L^{1}(\partial^{2}\times\Omega,m\times\omega)$ and $\bar{h}=V(h)\circ v$. We
will show that for $\mu\times\omega$-almost every $(\eta,w)$ there exists a
$\mu$-full measure subset such that if $\xi,\xi^{\prime}$ are in this subset
then:

$(\xi^{\prime},\eta),(\xi,\eta)\in
K\implies|f(\xi,\eta,w)-f(\xi^{\prime},\eta,w)|<\delta\cdot\bar{h}(\xi,\eta,w)$
(9)

By theorem 10.1 there exists a full measure subset $A$ of
$\partial^{2}G\times\Omega$ such that for all $(\xi,\eta,w)\in A$ and any
$a\in\mathbb{R}$:

$\lim_{T\to\infty}J_{a}^{a+T}(f)(\xi,\eta,w)=\bar{f}(\xi,\eta,w)$

$\lim_{T\to\infty}J_{a}^{a+T}(h)(\xi,\eta,w)=\bar{h}(\xi,\eta,w)$

By Fubini’s theorem it is enough to prove implication 9 for
$(\xi,\eta,w),(\xi^{\prime},\eta,w)\in A$.

Fix $\xi,\xi^{\prime},\eta,w$ such that $(\xi,\eta,w),(\xi^{\prime},\eta,w)\in
A$. By lemma 10.4 and using the uniform continuity of $\phi$ there exists
$0<a_{1}$ such that for all $g$ with $\sigma(g,\eta)>a_{1}$, if
$(\xi,\eta),(\xi^{\prime},\eta),(g\xi,g\eta)\in K$ then
$|\phi(g\xi,g\eta)-\phi(g\xi^{\prime},g\eta)|<\delta$, so:

$|f(g\xi,g\eta,gw)-f(g\xi^{\prime},g\eta,gw)|<\delta\cdot h(g\xi,g\eta,gw)$

On the other hand it follows from lemma 10.4 (after replacing the roles of
$\xi$ and $\xi^{\prime}$) and the fact that $supp(\phi)\subset int(K)$, that
there exists $0<a_{2}$ such that for all $g$ with $\sigma(g,\eta)>a_{2}$ if
$(\xi,\eta),(\xi^{\prime},\eta)\in K$ but $(g\xi,g\eta)\notin K$ then
$(g\xi,g\eta)(g\xi^{\prime},g\eta)\notin supp(\phi)$. Therefore:

$|f(g\xi,g\eta,gw)-f(g\xi^{\prime},g\eta,gw)|=\delta\cdot h(g\xi,g\eta,gw)=0$

So if $a=max\\{a_{1},a_{2}\\}$ then for any $g$ with $\sigma(g,\eta)>a$, if
$(\xi,\eta),(\xi^{\prime},\eta)\in K$ then:

$|f(g\xi,g\eta,gw)-f(g\xi^{\prime},g\eta,gw)|<\delta\cdot h(g\xi,g\eta,gw)$

Now if $(\xi,\eta),(\xi^{\prime},\eta)\in K$ then:

$\displaystyle|\bar{f}(\xi,\eta,w)-\bar{f}(\xi^{\prime},\eta,w)|=|\lim_{T\to\infty}J_{a}^{a+T}(f)(\xi,\eta,w)-\lim_{T\to\infty}J_{a}^{a+T}(f)(\xi^{\prime},\eta,w)|$
$\displaystyle\leq$ $\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{\\{g\in
G|\sigma(g,\eta)\in[a,a+T]\\}}|f(g\xi,g\eta,gw)-f(g\xi^{\prime},g\eta,gw)|d\lambda(g)$
$\displaystyle\leq$ $\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{\\{g\in
G|\sigma(g,\eta)\in[a,a+T]\\}}\delta\cdot h(g\xi,g\eta,gw)d\lambda(g)$
$\displaystyle=$
$\displaystyle\delta\cdot\lim_{T\to\infty}J_{a}^{a+T}(h)(\xi,\eta,w)=\delta\cdot\bar{h}(\xi,\eta,w)$

Showing implication 9 as needed.

Since $K$ and $\delta$ are general we can take a sequence $K_{n}$ exhausting
$\partial^{2}G$ and $\delta_{n}\to 0$. Applying implication 9 to these
sequences we conclude that for $\mu\times\omega$ almost every $(\eta,w)$ there
exists a $\mu$ full measure subset such that if $\xi,\xi^{\prime}$ are in this
subset then:

$\bar{f}(\xi,\eta,w)=\bar{f}(\xi^{\prime},\eta,w)$

By symmetry the same statement holds reversing the roles of $\xi$ and $\eta$.

Using Fubini’s theorem we conclude that for $\omega$-almost every $w$ there
exist $\mu$-full measure subsets $A,B\subseteq\partial G$ such that for
$\xi\in A$, $\bar{f}(\xi,\cdot,w)$ is an essentially constant function of
$\eta$ and for $\eta\in B$, $\bar{f}(\cdot,\eta,w)$ is an essentially constant
function of $\xi$. Therefore for $\omega$-almost every $w$,
$\bar{f}(\cdot,\cdot,w)$ is essentially constant and thus $\bar{f}$ depends
only on the $\Omega$ coordinate. Since $\bar{f}$ is $G$-invariant and $G$ acts
ergodicaly on $(\Omega,\omega)$ we conclude $\bar{f}$ is essentially constant
as needed.

∎

Having proved ergodicity we know that the space $Y$ is a single point and the
operator $V$ is simply given by $V(f)=\int f(\xi,\eta,w)dm\times\omega$. We
can therefore re-state theorem 10.1 as follows:

######  Theorem 10.6.

For any $f\in L^{1}(\partial^{2}G\times\Omega,m\times\omega)$, for almost
every $(\xi,\eta,w)$ the values $I_{a}^{b}(f)(\xi,\eta,w)$ and
$J_{a}^{b}(f)(\xi,\eta,w)$ are finite for any interval $[a,b]$ and:

$\lim_{T\to\infty}I_{a}^{b+T}(f)(\xi,\eta,w)=\int f(\xi,\eta,w)dm\times\omega$

If in addition $f$ is supported on a bounded set then:

$\lim_{T\to\infty}J_{a}^{b+T}(f)(\xi,\eta,w)=\int f(\xi,\eta,w)dm\times\omega$

##  Appendix A Cocycles and Almost Cocycles

Let $G$ be a locally compact second countable group with a measure class
preserving action on a standard Lebesgue space $(X,\mu)$. A Borel cocycle of
the action $G\operatorname{\curvearrowright}X$ with values in $\mathbb{R}$ is
a Borel function $\alpha:G\times X\to\mathbb{R}$ such that for every $g,h\in
G$, for almost every $x\in X$:

$\alpha(gh,x)=\alpha(g,hx)+\alpha(h,x)$

A cocycle is called strict if the above equation holds for every $x$. Two
cocycles $\alpha,\rho$ are called cohomologous if there exists a function
$F:X\to\mathbb{R}$ such that for every $g$, for almost every $x$:

$\alpha(g,x)-\rho(g,x)=F(gx)-F(x)$

If $\alpha,\rho$ are strict cocycles they are strictly cohomologous if the
above equation holds for every $x$.

A Borel almost cocycle of the action $G\operatorname{\curvearrowright}X$ with
values in $\mathbb{R}$ is a Borel function $\sigma:G\times X\to\mathbb{R}$
such that for every $g,h\in G$, for almost every $x\in X$:

$\sigma(gh,x)\approx\sigma(g,hx)+\sigma(h,x)$

An almost cocycle is called strict if the above equation holds for every $x$.

######  Lemma A.1.

Let $G$ be a second countable locally compact group with a measure class
preserving action on a standard Lebesgue space $(X,\mu)$. Suppose that
$\alpha,\sigma:G\times X:\to\mathbb{R}$ are a strict Borel cocycle and a
strict Borel almost cocycle respectively such that for every $g\in G$, for
almost every $x\in X$, $\alpha(g,x)\approx\sigma(g,x)$. There exists a strict
cocycle $\rho$ strictly cohomologous to $\alpha$ and a $G$-invariant full
measure Borel set $S\subseteq X$ such that for all $(g,x)\in G\times S$,
$\rho(g,x)\approx\sigma(g,x)$.

###### Proof.

For $x\in X$ denote $E_{x}(C)=\\{g\in G||\alpha(g,x)-\sigma(g,x)|<C\\}$. For
every $g\in G$, for almost every $x\in X$, $\alpha(g,x)\approx\sigma(g,x)$ so
by Fubini’s theorem there exists a full measure Borel set $A\subseteq X$ and
$C_{1}>0$ such that for every $x\in A$, $E_{x}(C_{1})$ has full Haar measure.
By [22, Appendix B.8] there exists a full measure Borel subset $B\subseteq A$
such that $S=G.B$ is Borel. If $E_{x}(C_{1})$ has full Haar measure then there
exists $C_{2}>0$ such that for almost every $g\in G$, $E_{gx}(C_{2})$ has full
Haar measure. Indeed if $g\in E_{x}(C_{1})$ then for any $h\in
E_{x}(C_{1})g^{-1}$:

$\alpha(h,gx)=\alpha(hg,x)-\alpha(g,x)\approx\sigma(hg,x)-\sigma(g,x)\approx\sigma(h,gx)$

Since every $x\in S$ has an orbit which intersects $B$ we conclude that for
any $x\in S$, for almost all $g\in G$, $E_{gx}(C_{2})$ has full Haar measure.

We claim that there exists a Borel function $F:S\to\mathbb{R}$ and $C_{3}>0$
such that for every $x\in S$, for almost every $g\in G$,
$|\alpha(g,x)-\sigma(g,x)-F(x)|<C_{3}$. Note that for almost every $x\in B$ we