# Spectral gaps, critical exponents and representations of negatively curved
groups

###### Abstract.

In this paper we introduce a notion of Poincaré exponent for isometric
representations of discrete groups on Hilbert spaces. Similarly as growth
exponents control the geometry this exponent is shown to control the size of
spectral gaps. Following similar ideas as Patterson and Sullivan in [71] [78]
it is used in the case of negatively curved groups to construct weakly
contained boundary representations [8] reflecting the spectral properties of
the original representation analogously as complementary series
representations in the case of semi-simple Lie groups [77]. This is exploited
to deduced sharp estimates on spectral invariants. A quantitive property (T) á
la Cowling [32] is also established proving uniform bound on the mixing rate
of representations of hyperbolic groups with property (T). Along the way some
properties of boundary representations are discussed. A original
characterisation of the positivity of the so-called Knapp-Stein operators and
certain fusion rules on the boundary complementary series representations [15]
([22]) are established.

KEVIN BOUCHER

University of Southampton, UK

(e-mail<EMAIL_ADDRESS>
Key words: Poincaré series, Patterson-Sullivan theory, hyperbolic groups,
complementary series, boundary representations, mixing rates, spectral gap.
2020 Mathematics Subject Classification: 20F67 22D10 37A46 43A35

###### Contents

  1. 1 Introduction
    1. 1.1 Boundary representations
    2. 1.2 A example: homogeneous dynamics of $SL_{2}({\mathbb{R}})$
    3. 1.3 Statement of results
      1. 1.3.1 Critical exponent
      2. 1.3.2 Critical exponent and negatively curved groups
      3. 1.3.3 Critical exponent and boundary representations
    4. 1.4 Organisation of the paper
    5. 1.5 Notations and symbols
    6. 1.6 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Strong hyperbolicity and consequences
    2. 2.2 Orbital distribution
    3. 2.3 $s$-density representations and summation estimates
    4. 2.4 Nets and positive families of vectors
    5. 2.5 Observation about the GNS of pointwise positive functions
  3. 3 Boundary representations of positive cyclic representations
    1. 3.1 Critical exponent and Poincaré series
    2. 3.2 A revisited Gelfand-Naimark-Segal construction
    3. 3.3 A measure theoretic perspective
    4. 3.4 Back to quadratic forms
  4. 4 First properties
    1. 4.1 The relations between $\pi_{\phi}^{\partial}$ and $\pi_{\phi}$
      1. 4.1.1 About the boundary representation of a boundary representation
    2. 4.2 Natural complementary series
    3. 4.3 About the critical case $\delta=\delta(\phi)$
    4. 4.4 About the regular representation
  5. 5 Spectral bounds
    1. 5.1 Rapid decay and critical exponent
    2. 5.2 Spectral gap and exponential mixing
    3. 5.3 Quantitative property (T) for hyperbolic groups
    4. 5.4 Spectral gap and boundary representations
  6. A Norm transfer estimate
  7. B Weak operator topology

##  1\. Introduction

Given a topological group, $G$, together with a continuous unitary
representation on a Hilbert space, $(\pi,\mathcal{H})$, quantitative estimates
on the operator norms $\|\pi(\mu)\|_{\text{op}}$ for Borel probability
measures, $\mu$, (see [37] for technical details) is a classic problem in
spectral methods involving $G$-representations. The representation $\pi$ of
$G$ is said to have spectral gap if there exists a Borel probability measure,
$\mu$, with $\overline{\langle\text{supp}(\mu)\rangle}=G$ such that
$\|\pi(\mu)\|_{\text{op}}<1$ (see [9, Prop. 3.4.] for other
characterisations). Given a Borel set $A\subset G$ and denoting
$\text{Avr}_{A}(\pi)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{\text{Vol}(A)}\int_{A}\pi(g)dg$,
a central question is to obtain a bound on
$\|\text{Avr}_{A}(\pi)\|_{\text{op}}$ in terms of the volume of $A$, in other
words understand how the spectral gap evolves when $A$ grows in volume. This
has a number of applications going from homogeneous dynamic [9] to quantum
ergodicity [61] passing through number theory [45], counting problems [44],
Apollonian circle packing problems [57] or random walk on groups [81].

If $G$ is a semi-simple Lie group it follows from a deep machinery involving
classification of irreducible unitary representations together with matrix
coefficient estimates that precise quantitative estimates on spectral gaps can
be obtained for bi-$K$-invariant balls [67] where $K\subset G$ stands for a
maximal compact subgroup of $G$.

In this present work we address the situation of discrete negatively curved
groups in a broad sense by setting the role of boundary complementary series
representations within the problem of spectral gap estimates. Ergodic theorems
for measure preserving action of negatively curved groups actions has a long
history [25] [40] [18] [19] [20] but, as far as the author knows, no
quantitive description of the spectral gap have been established even in this
particular case.

###  1.1. Boundary representations

The subject of boundary representations of discrete groups popularised by
Bader and Muchnik in [8] can be traced back to the work of Figa-Talamanca and
Piccardello on free groups [39]. Bader and Muchnik investigated the
irreducibility of the Koopman representations of negatively curved groups on
their visual boundaries endowed with their Patterson-Sullivan measures from a
dynamical perspective. First established by them in the case of fundamental
groups of negatively curved manifolds, it was latter extended to CAT(-1)
groups by Boyer [21] and Gromov hyperbolic groups by Garncarek [41]. Those
representations recently inspired different work such as classification of
type 1 topological hyperbolic groups and their boundary representations in
[26] and [43], classification of additive representations of free groups in
relation with spectral gap problems in [74] or boundary representation of
Mapping class groups [62]. Following the analogy between hyperbolic groups and
uniform lattices of rank 1 semi-simple Lie groups the author introduced a
notion of spherical complementary series representations in [15] (see also
[22]). Those representations, when they exist, produce a non-tempered family
of non-equivalent, irreducible, unitary representations with intermediate
spectral gap properties. Their existence relates to properties such as the
Kazhdan property (T) and its natural obstruction namely the Haagerup property
[15] [16]. They also showed remarkable approximation properties in relation
with the so-called Shalom conjecture [17], [5], [69], [46] and [70, Conj. 35]
for context.

In the semi-simple case, the relation between complementary series
representations and decay of matrix coefficients [77] are fundamental in order
to investigate spectral gaps. Our approach is to extend this relation using
complementary boundary representations, as a substitute to the semi-simple
ones.

###  1.2. A example: homogeneous dynamics of $SL_{2}({\mathbb{R}})$

Let us illustrate our point through the example of $SL_{2}({\mathbb{R}})$
viewed as a topological Gromov hyperbolic group. The reader can refer to [55]
for material about $SL_{2}({\mathbb{R}})$ representation theory or [6, Sect.
3.4] for a gentle summary.

Let $\Gamma\subset SL_{2}({\mathbb{R}})$ be a discrete, finitely generated and
not virtually solvable subgroup (see [56] for examples) and
$L^{2}(\Gamma\setminus SL_{2}({\mathbb{R}}))$ be the quasi-regular
representation of $SL_{2}({\mathbb{R}})$ on square integrable
$\Gamma$-automorphic functions on $SL_{2}({\mathbb{R}})$. Assume the critical
exponent $\delta_{\Gamma}$ of $\Gamma$ with respect to the canonical distance
on $SL_{2}({\mathbb{R}})$ satisfies $\delta_{\Gamma}>\frac{1}{2}$.
It follows from Lax and Phillips [60] that:

$L^{2}(\Gamma\setminus SL_{2}({\mathbb{R}}))\simeq
W_{\lambda_{0}}\oplus\dots\oplus W_{\lambda_{m}}\oplus W_{\text{temp.}}$

where $W_{\lambda_{i}}$ is a spherical complementary series of Langland
parameter $0<s_{i}\leq 1$ ($s=1$ corresponds to the trivial representation)
and $W_{\text{temp.}}$ the tempered spectrum, i.e a
$SL_{2}({\mathbb{R}})$-representation with $L^{2+\varepsilon}$-integrable
matrix coefficients [31]. Each spherical complementary series representation
$W_{\lambda_{i}}$ with $0\leq i\leq m$ is generated by an unique
$SO(2)$-invariant $\lambda_{i}=s_{i}(1-s_{i})$-eigenvector of the hyperbolic
Laplace operator on $L^{2}(\Gamma\setminus SL_{2}({\mathbb{R}}))^{SO(2)}\simeq
L^{2}(\Gamma\setminus\mathbb{H}^{2})$ where $0\leq\lambda_{i}\leq\frac{1}{4}$.
Moreover, due to Sullivan’s theorem [78], one has
$\lambda_{0}=\delta(1-\delta)$. In particular only finitely many spherical
complementary series appear in $L^{2}(\Gamma\setminus SL_{2}({\mathbb{R}}))$.

######  Remark 1.1.

This decomposition has its generalisation in higher dimension with
$\Gamma\subset SO(n+1,1)$ discrete, geometrically finite with critical
exponent $\delta>\frac{n}{2}$ and Zariski dense (see [66] and references
therein).

The above example illustrates the general fact that spherical complementary
series control the decay of matrix coefficients (see [77]) and thus spectral
gap of average operators such as $\frac{1}{\text{Vol}(A)}\int_{A}\pi(g)dg$ for
$A\subset SL_{2}({\mathbb{R}})$ and $\pi$ a
$SL_{2}({\mathbb{R}})$-representation (using spectral transfer [67] [9]).

Sullivan’s theorem guarantees that $L^{2}(\Gamma\setminus
SL_{2}({\mathbb{R}}))$ contains no complementary series for $\delta<s\leq 1$
and therefore a control of the mixing rate for the representation in terms of
$\delta$. This is crucial to deduce the correct normalisation of correlation
functions in matrix coefficients equidistribution formulas such as in [75] or
[66, Thm. 1.4] and also play a role in the study of boundary representations
[8], [21],[41],[15], [22].

A major difficulty while considering discrete groups is that one cannot rely
on integrability properties of matrix coefficients to approach the spectral
gap of a given representation as decay of matrix coefficients and spectral gap
turn out to be distinct notions:

######  Example 1.2.

Let $\Gamma$ be a discrete group and $\Gamma^{\prime}\lhd\Gamma$ an infinite
normal subgroup with non-amenable quotient $\Gamma/{\Gamma^{\prime}}$. The
quasi-regular representation
$(\lambda_{\Gamma^{\prime}},\ell^{2}(\Gamma/{\Gamma^{\prime}}))$ is an example
of $\Gamma$-representation with spectral gap (due to Kesten [53]) but no
matrix coefficient in $\ell^{p}(\Gamma)$ for any $p\neq\infty$ nor even if in
$c_{0}(\Gamma)$.

Following ideas of Patterson and Sullivan [71] [78] on their investigation of
the geometry and dynamic related to Kleinian groups, the notion of
integrability is substituted with a representation theoretic notion of
critical exponent:

###  1.3. Statement of results

####  1.3.1. Critical exponent

Let $\Gamma$ be a discrete finitely generated group endowed with a
$\Gamma$-invariant distance, $d_{\Gamma}$, quasi-isometric to a word distance
and $\delta$ its critical exponent defined as
$\delta\stackrel{{\scriptstyle\text{def}}}{{=}}\limsup_{r}\frac{1}{r}\log|B_{r}|$
where $B_{r}$ stands for a $d_{\Gamma}$-ball of radius $r$ in $\Gamma$.

A unitary representation $(\pi,\mathcal{H})$ together with a unit vector
$v\in\mathcal{H}$ such that
$\overline{\text{Span}\\{\pi(g)v:g\in\Gamma\\}}=\mathcal{H}$ is called cyclic.
Two cyclic representations $(\pi,v)$ and $(\pi^{\prime},v^{\prime})$ are
equivalent if there exists an isometric intertwiner, $U$, between $\pi$ and
$\pi^{\prime}$ such that $U(v)=v^{\prime}$. The classes of such
representations are in bijective correspondence with positive definite
functions on $\Gamma$ (see Subsection 2.5 or [10]) through the map
$[\pi,v]\leftrightarrow\phi_{\pi,v}(.)\stackrel{{\scriptstyle\text{def}}}{{=}}(\pi(.)v,v)\in{\mathbb{C}}^{\Gamma}$.
We shall use the identification between positive definite functions on
$\Gamma$ and cyclic representations without explicit mention throughout the
rest.
A cyclic representation $[\pi,v]$ is called positive if $(\pi(.)v,v)\geq 0$ on
$\Gamma$ and we denote the set of classes of positive cyclic unitary
representations $\mathbb{P}_{+}(\Gamma)$.

######  Example 1.3 (Positive cyclic representations).

  1. (1)

The trivial representation associated to the constant function, ${\bf
1}_{\Gamma}$, on $\Gamma$;

  2. (2)

Given a subgroup $H\subset\Gamma$, $[\lambda_{H},[\text{Dir}_{e}]]$ where
$\lambda_{H}$ stands for the left quasi-regular representation of $\Gamma$ on
$\ell^{2}(\Gamma/H)$ and $[\text{Dir}_{e}]=\text{Dir}_{H}$ the class of
$e\in\Gamma$ in $\Gamma/H$;

  3. (3)

Given a unitary cocycle $b:\Gamma\rightarrow\mathcal{H}$, that is a map that
satisfies $b(gh)=b(g)+\pi(g)b(h)$ for all $g,h\in\Gamma$ and some unitary
representation $(\pi,\mathcal{H})$ (see [11, p.76] for details), then
$\phi(g)=e^{-t\|b(g)\|^{2}}\in\mathbb{P}_{+}(\Gamma)$ for all
$t\in{\mathbb{R}}_{+}$;

  4. (4)

Given measure class preserving action $\Gamma\curvearrowright(\Omega,m)$ and
$u\in L^{2}_{+}(\Omega)$ with $\|u\|_{2}=1$,
$\phi(g)=(\pi_{\text{Koop}}(g)u,u)\in\mathbb{P}_{+}(\Gamma)$ where
$\pi_{\text{Koop}}$ stands for the $L^{2}$ Koopman representation:

$\pi_{\text{Koop}}(g)f(\omega)=\frac{dg_{*}m}{dm}(\omega)f(g^{-1}\omega)$

for $m$-almost every $\omega\in\Omega$;

  5. (5)

The real boundary representations, $\pi_{s}$, [8] [41] [15] together with the
constant vector ${\bf 1}_{{\partial}\Gamma}$ also belong to
$\mathbb{P}_{+}(\Gamma)$.

More generally, given a unitary representation $(\pi,\mathcal{H})$ of $\Gamma$
and a unit vector $v\in\mathcal{H}$ one can consider the cyclic representation
$[\pi\otimes\overline{\pi}|_{W_{v}},v\otimes\overline{v}]$ where
$\overline{\pi}$ stands for the contragredient representation and
$W_{v}=\overline{\text{Span}\\{\pi(g)\otimes\overline{\pi(g)}(v\otimes\overline{v}):g\in\Gamma\\}}\subset\mathcal{H}\otimes\overline{\mathcal{H}}$.
Observe that $[\pi\otimes\overline{\pi}|_{W_{v}},v\otimes\overline{v}]$ can be
seen as a positive representation as
$(\pi\otimes\overline{\pi}(v\otimes\overline{v}),v\otimes\overline{v})=|(\pi
v,v)|^{2}$.

The critical exponent of a positive cyclic representation $[\pi,v]$ is defined
as the abscissa of convergence of the weighted Poincaré series:

$\mathcal{P}([\pi,v];t)=\mathcal{P}((\pi
v,v);t)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{g\in\Gamma}e^{-td_{\Gamma}(g,e)}(\pi(g)v,v)$

and denoted $\delta[\pi,v]$ (or $\delta(\phi)$ for $\phi=(\pi v,v)$). Observe
that $\delta[\pi,v]\leq\delta$.

######  Remark 1.4.

In relation with Brook’s amenable cover problem a notion of critical exponent
for measure preserving transformation was introduced in [30] in order to
investigate the spectral properties of $\ell^{2}(\Gamma/{\Gamma^{\prime}})$ in
the particular case where $\delta_{\Gamma}=\delta_{\Gamma^{\prime}}$ and
$\Gamma$ with some form of hyperbolicity (see [30] for details). As
$[\lambda_{\Gamma^{\prime}},[\text{Dir}_{e}]]$ is a positive cyclic
representation one can compare the two notions and show that
$\delta[\lambda_{\Gamma^{\prime}},[\text{Dir}_{e}]]=\delta_{\Gamma^{\prime}}\leq\delta(\lambda_{\Gamma^{\prime}})$
where $\delta_{\Gamma^{\prime}}$ stands for the critical exponent of
${\Gamma^{\prime}}$ in $(\Gamma,d_{\Gamma})$ and
$\delta(\lambda_{\Gamma^{\prime}})$ the critical exponent of
$\lambda_{\Gamma^{\prime}}$ in the sense of [30].

If $\lambda$ stands for the regular representation of $\Gamma$ on
$\ell^{2}(\Gamma)$ the following spectral inequality holds:

######  Theorem 1.

Let $[\pi,v]$ be a positive cyclic representation. Then for all
$f\in{\mathbb{C}}[\Gamma]$ one has:

$\|\pi(f)\|_{\text{op}}\leq
e^{\frac{1}{2}\delta[\pi,v]r(f)}\|\lambda(f)\|_{\text{op}}$

where $r(f)$ stands for the smallest $r$ such that $\text{supp}(f)\subset
B_{r}$ the $d_{\Gamma}$-ball of radius $r$ around $e\in\Gamma$.

As observed by Shalom in [77, Lem. 2.3]: every positive cyclic representation
$[\pi,v]$ satisfies $\|\lambda(f)\|_{\text{op}}\leq\|\pi(f)\|_{\text{op}}$ for
all $f\in{\mathbb{R}}_{+}[\Gamma]$. It is therefore natural to have a spectral
estimate that involves the regular representation.

A function $f\in{\mathbb{C}}[\Gamma]$ on $\Gamma$ is radial if
$f(g)=k(d_{\Gamma}(g,e))$ for some compactly supported function $k$ on
${\mathbb{R}}$.

######  Definition ([28]).

The discrete metric group $(\Gamma,d_{\Gamma})$ has the radial rapid decay
(RRD) property, if there exist constants $C\in{\mathbb{R}}_{+}$ and
$m\in{\mathbb{N}}$ such that for all radial function
$f\in{\mathbb{C}}[\Gamma]$ supported in a ball of radius $R$:
$\|\lambda(f)\|_{\text{op}}\leq CR^{m}\|f\|_{2}$.

This property is satisfied by Gromov hyperbolic groups [50], CAT(0) cubical
groups [27], cocompact lattices of connected semi-simple Lie groups [72] and
others [28]. It also appears to be strictly weaker than the usual rapid decay
[23] popularised by Lafforgue in [59].

Let $B_{r}$ be the ball of radius $r$ around the unit element $e\in\Gamma$ and
denote
$\text{Avr}_{r}(\pi)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{|B_{r}|}\sum_{g\in
B_{r}}\pi(g)$. The asymptotic behaviour of
$\|\text{Avr}_{r}(\pi)\|_{\text{op}}$ is independent of the centre of $B_{r}$
and the entropy (or mixing rate) of $\pi$ is defined as:

$h(\pi)\stackrel{{\scriptstyle\text{def}}}{{=}}-\limsup_{r}\frac{1}{r}\log\|\text{Avr}_{r}(\pi)\|_{\text{op}}\geq
0$

######  Remark 1.5.

Let $\text{Rep}(\Gamma)$ be the space of separable unitary
$\Gamma$-representations. The map $\pi\in\text{Rep}(\Gamma)\mapsto
h(\pi)\in{\mathbb{R}}_{+}$ is monotone with respect to weak containment [11,
Appendix F]. More precisely if $\pi,\pi^{\prime}\in\text{Rep}(\Gamma)$ with
$\pi\prec\pi^{\prime}$ then $h(\pi^{\prime})\leq h(\pi)$ with equality
whenever $\pi$ and $\pi^{\prime}$ are Fell equivalent [11, Thm. F.4.4].

The following result establishes a general relation between the entropy and
critical exponent:

######  Corollary 1.6.

If $\Gamma$ has the radial rapid decay property, there exists
$m\in{\mathbb{N}}$, for all $\varepsilon>0$, there exist
$C(\varepsilon),R_{\varepsilon}>0$ such that

$\|\text{Avr}_{r}(\pi)\|_{\text{op}}\leq
C(\varepsilon)r^{m}e^{\frac{1}{2}(\delta(\pi)-\delta-\varepsilon)r}$

for all $r\geq R_{\varepsilon}$. In particular
$h(\pi)\geq\frac{1}{2}(\delta-\delta[\pi,v])$.

######  Corollary.

If $\Gamma$ has the radial rapid decay property and
$\Gamma^{\prime}\subset\Gamma$ a subgroup with
$\delta_{\Gamma^{\prime}}<\delta_{\Gamma}$, then mixing rate of
$\ell^{2}(\Gamma/{\Gamma^{\prime}})$ is strictly positive.

A unitary representation, $(\pi,\mathcal{H})$, is called strongly
$L^{p+\varepsilon}$ with $p\neq\infty$ if for all $p^{\prime}>p$, there exists
a dense subspace, $\mathcal{D}_{p^{\prime}}\subset\mathcal{H}$, such that the
matrix coefficients $(\pi(.)v,w)\in\ell^{p^{\prime}}(\Gamma)$ for all
$v,w\in\mathcal{D}_{p^{\prime}}$.

######  Theorem 2.

Let $\pi$ be a strongly $L^{p+\varepsilon}$ unitary representation with $2\leq
p\neq\infty$, then for all $f\in{\mathbb{C}}[\Gamma]$:

$\|\pi(f)\|_{\text{op}}\leq e^{\frac{p-2}{2p}\delta
r(f)}\|\lambda(f)\|_{\text{op}}.$

In particular if $(\Gamma,d_{\Gamma})$ has the radial rapid decay property one
has $h(\pi)\geq\frac{1}{p}\delta$.

######  Remark 1.7.

It follows from [67, Thm. 1] and [77, Lem. 2.3] that
$\|\pi(f)\|_{\text{op}}=\|\lambda(f)\|_{\text{op}}$ for all
$f\in{\mathbb{R}}_{+}[\Gamma]$ whenever $[\pi,v]$ is positive and strongly
$L^{p+\varepsilon}$ with $1\leq p\leq 2$.

######  Remark 1.8.

It was known by spectral transfer argument (see [67, Thm. 1]) that
$\|\text{Avr}_{r}(\pi)\|_{\text{op}}\leq Ce^{-\beta r}$ for some $\beta(p)>0$
whenever $\pi$ is strongly $L^{p}$ and $(\Gamma,d_{\Gamma})$ has the radial
rapid decay. As far as the author knows such precise bound is new in this
general context.

Observe that Theorem 2 shows some deficiency in Theorem 1. Indeed assuming the
Poincaré series $t\mapsto\sum_{g\in\Gamma}e^{-td_{\Gamma}(g,e)}$ diverges at
$\delta$, i.e $\Gamma$ is of divergent type, the functions
$f_{s}(g)=e^{-(1-s)\delta d_{\Gamma}(g,e)}$ for $\frac{1}{2}<s<1$ belong to
$\ell^{\frac{1}{1-s}+\varepsilon}(\Gamma)\setminus\ell^{\frac{1}{1-s}}(\Gamma)$
with $p=\frac{1}{1-s}>2$ and satisfy $\delta(f_{s})=s\delta$. If $f_{s}$ (or
$\phi_{s}\asymp f_{s}$ with $\phi_{s}\in\mathbb{P}_{+}(\Gamma)$) is positive
definite, Theorem 2 gives a sharper estimate of the mixing rate than Theorem 1
(see Subsection 5.2 for more details). This suggest the better estimate on
mixing rate in terms of the critical exponent :
$h(\pi)\geq\delta-\delta[\pi,v]$.

As discussed in the next subsection this sharper bound is actually optimal
under the assumption that $\Gamma$ is hyperbolic.

####  1.3.2. Critical exponent and negatively curved groups

Assume for the rest of this introduction that $\Gamma$ is a non-elementary
Gromov hyperbolic group and $d_{\Gamma}$ is strongly hyperbolic (see
Definition 2.1).

######  Remark 1.9.

Due to Mineyev and Yu in [65] (see also [64] and [68]) every Gromov hyperbolic
group acts geometrically on a proper roughly geodesic strongly hyperbolic
space.

######  Theorem 3.

Let $[\pi,v]$ be a positive cyclic representation with critical exponent
$\delta[\pi,v]$. Then

$h(\pi)=-\lim_{r}\frac{1}{r}\log\|\text{Avr}_{r}(\pi)\|_{\text{op}}=\delta-\max\\{\delta[\pi,v],\frac{1}{2}\delta\\}$

In particular $0\leq h(\pi)\leq\frac{\delta}{2}$ and $\delta[\pi,v]$ is
independent of the choice of the positive cyclic vector $v$ and can be denoted
$\delta(\pi)$.

A careful analysis of the critical case $\delta=\delta(\pi)$ leads to:

######  Theorem 4.

Let $[\pi,v]$ be a positive cyclic representation with critical exponent
$\delta(\pi)$. The following are equivalent:

  1. (1)

$\pi$ has almost invariant vectors;

  2. (2)

$\delta(\pi)=\delta$, i.e $[\pi,v]$ has no critical gap;

  3. (3)

$\pi$ has trivial entropy, i.e $h(\pi)=0$.

This result can be seen as a generalisation of [30, Thm. 1] with
$[\pi,v]=[\lambda_{{\Gamma^{\prime}}},[\text{Dir}_{e}]]$.

A representation, $(\pi,\mathcal{H})$, is called weakly mixing if it does not
contain any finite dimension subrepresentation or equivalently
$(\pi\otimes\overline{\pi},\mathcal{H}\otimes\overline{\mathcal{H}})$ does not
have non-zero invariant vectors [11, Prop A.1.12]. Similarly as Cowling’s
quantitative property (T) [32] for semi-simple Lie groups, hyperbolic groups
with Kazhdan property (T) [11] have a uniform lower bound on the entropy of
their mixing representations:

######  Theorem 5.

If $\Gamma$ has the property (T) then there exists
$\beta(\Gamma,d_{\Gamma})>0$, for all mixing representation, $\pi$, one has
$h(\pi)\geq\beta(\Gamma,d_{\Gamma})$.

The main ingredient in our approach is the existence of a critical weakly
contained boundary representation of conformal order $\delta(\pi)$ :

####  1.3.3. Critical exponent and boundary representations

Let ${\partial}\Gamma$ be the Gromov boundary of $\Gamma$, the Gromov product
based at $e\in\Gamma$ between $\xi,\eta\in{\partial}\Gamma$ is denoted
$(\xi,\eta)\in[0,\infty]$ and $b_{x}$ the Buseman cocycle at
$x\in\Gamma\cup{\partial}\Gamma$ (see Subsection 2.1). Let $\pi_{s}$ be the
linear representation of $\Gamma$ on ${\mathcal{C}}(\overline{\Gamma})$ given
by

$\pi_{s}(g)\psi(x)=e^{-s\delta b_{x}(g,e)}\psi(g^{-1}x)$

for all $g\in\Gamma$ and $x\in\overline{\Gamma}$.

The analogue of spherical complementary series are determined by the (bounded)
$s$-Knapp-Stein operators, $\mathcal{I}_{s}$, for $\frac{1}{2}<s<1$ defined on
$L^{2}(\nu_{\text{PS}})$ as:

$\mathcal{I}_{s}(\psi)(\xi)=\int_{{\partial}\Gamma}\psi(\eta)e^{2(1-s)\delta(\xi,\eta)}d\nu_{\text{PS}}(\eta)$

where $\nu_{\text{PS}}$ stands for the Patterson-Sullivan measure. The
$s$-spherical complementary series (also called natural complementary series
in the rest) are defined whenever $\mathcal{I}_{s}$ is positive definite and
are obtained from the completion on $L^{2}(\nu_{\text{PS}})$ with respect to
the $\pi_{s}$-invariant quadratic form:

$Q_{s}(\psi)=(\mathcal{I}_{s}(\psi),\psi)_{L^{2}(\nu_{\text{PS}})}$

see [15] ([22]) for more details and [39, Chap. 5] for the case of free
groups.

As non virtually abelian discrete groups are not type $1$ [10, Chap. 7] their
unitary dual cannot be canonically understood. This is reflected on the
boundary through the existence of boundary representation of exotic type that
does not have analogue within the semi-simple case and motivate the following
definition:

Given $\nu$ a $\Gamma$-quasi-invariant probability measure over the boundary
${\partial}\Gamma$, a conditional expectation, $E$, on $L^{1}(\nu)$ is called
$(\Gamma,s,\nu)$-conformal if it satisfies $\pi^{*}_{s}(g)E=E\pi_{s}(g)$ for
all $g\in\Gamma$ on $L^{2}(\nu)\subset L^{1}(\nu)$, where $\pi^{*}_{s}(g)$
stands for the adjoint of $\pi_{s}(g)$ on $L^{2}(\nu)$. Assuming the quadratic
form

$Q_{E}(f)\stackrel{{\scriptstyle\text{def}}}{{=}}\iint
f(\xi)E(f)(\xi)d\nu(\xi)$

associated to $E$ on $L^{2}(\nu)$ is positive definite it induces a
$\pi_{s}$-invariant scalar product and therefore a $\Gamma$-representation. A
boundary representation is called $s$-complementary series representation (or
$(\nu,E,s)$-complementary series to insist on the structure) if it is obtained
through the above procedure.

######  Theorem 6.

Let $(\Gamma,d_{\Gamma})$ be Gromov hyperbolic endowed with a strongly
hyperbolic distance as above and $[\pi,v]$ be a positive cyclic representation
with critical exponent $\delta([\pi,v])$. There exists a
$(\nu,E,s)$-complementary series representation, $\pi^{\partial}$, with
$s\delta=\delta[\pi,v]$ factorising through the kernel of $\pi$ that is weakly
contained in $\pi$. Moreover $\delta(\pi^{\partial})=s\delta=\delta[\pi,v]$
with $\pi^{\partial}=1$ whenever $\delta[\pi,v]=\delta$ and $\pi$ does not
weakly contain any boundary representation of conformal index strictly greater
than $\delta(\pi)$.

In certain situations the representations $\pi^{\partial}$ induce by Theorem 6
can be realised as an actual subrepresentation of $\pi$ or even be identified
with the natural complementary series induced by the Knapp-Stein operator:

######  Definition 1.10.

A positive function $f$ on $\Gamma$ is called Poincaré spherical if there
exist $\delta(f)<s_{0}$ and $C$ such that the inequality:

$\sum f(g^{-1}h)e^{-\sigma d_{\Gamma}(h,e)}e^{-\sigma d_{\Gamma}(g,e)}\leq
C.\mathcal{P}(f;\sigma)^{2}$

holds for all $\delta(f)<\sigma\leq s_{0}$.

See Subsection 4.1 for more explanations about this definition.

######  Proposition 1.11.

Let $\phi=(\pi v,v)\in\mathbb{P}_{+}(\Gamma)$ that is Poincaré spherical and
$[\pi,v]$ its positive cyclic representation. Then the space of intertwiners
between $\pi^{\partial}$ and $\pi$ is non-trivial. Moreover if
$\pi^{\partial}$ is irreducible then $\pi^{\partial}$ is a subrepresentation
of $\pi$.

######  Remark 1.12.

It is likely that any boundary representation obtained from the above
procedure must be irreducible.

Given a real positive function, $k$, a function $f$ on $\Gamma$ is called
$k$-roughly-spherical if $f\asymp k(d_{\Gamma}(\bullet,e))$. Let $k_{s}$ be
the real function $k_{s}(t)=\exp({-(1-s)\delta t})$ for $t\geq 0$. Then a
$k_{s}$-roughly-spherical function on $\Gamma$ is Poincaré spherical (see
Proposition 4.5) and the spectral properties of those type of functions relate
to the spectrum of the Knapp-Stein operators introduced before:

######  Theorem 7.

If a pointwise positive, positive definite function $\phi$ is $k_{s}$-roughly
radial then the Knapp-Stein operator $\mathcal{I}_{s}$ is positive definite on
$L^{2}(\nu_{\text{PS}})$, the boundary representation $\pi_{\phi}^{\partial}$
is uniquely defined and isomorphic to the natural complementary series
$\pi_{s}$. Moreover $\pi_{s}$ appears as a subrepresentation of $\pi_{\phi}$.

This leads to the following characterisation of positivity for the Knapp-Stein
operator $\mathcal{I}_{s}$ on $L^{2}(\nu_{\text{PS}})$:

######  Corollary 1.13.

Let $k_{s}$ be the real function $k_{s}(t)=\exp({-(1-s)\delta t})$. The
operator $\mathcal{I}_{s}$ on $L^{2}(\nu_{\text{PS}})$ is positive definite if
and only if $\Gamma$ has a $k_{s}$-roughly-spherical positive definite
function.

We call a unitary cocycle $b:\Gamma\rightarrow\mathcal{H}$ special if the
Hilbert pseudo-distance $d_{b}(g,h)=\|b(g^{-1}h)\|^{2}$ for $g,h\in\Gamma$ is
roughly isometric to $d_{\Gamma}$, i.e $\|d_{\Gamma}-c.d_{b}\|_{\infty}\leq C$
for some finite $C$ and $c>0$.

######  Corollary 1.14 (Fusion rules).

Let $\frac{1}{2}<s,s^{\prime},t\leq 1$ such that $s+s^{\prime}=1+t$ and
assuming $\pi_{s}$ and $\pi_{s^{\prime}}$ are well defined natural
complementary series, then $\pi_{t}$ is also well defined and
$\pi_{t}\subset\pi_{s}\otimes\pi_{s^{\prime}}$. Moreover if $\Gamma$ admits a
special cocycle $c$ then the natural complementary series
$(\pi_{s})_{s\in(\frac{1}{2},1]}$ exist and $\pi_{s}\subset
e^{-(1-s)\|c\|^{2}}$ for all $\frac{1}{2}<s\leq 1$.

###  1.4. Organisation of the paper

Section 2 starts with general material about negatively curved spaces and
groups. Estimates on spherical sums of the $s$-density representations are
discussed in Subsection 2.3. Elements of Banach geometry and their relations
with the so-called Gelfand-Naimark-Segal (or GNS) construction of positive
cyclic representations are respectively discussed in Subsections 2.4 and 2.5.

The section 3 is dedicated to the constructive part of this work, namely the
boundary representations of positive cyclic representations. After discussing
properties of Poincare series on negatively curved groups in Subsection 3.1,
the construction is performed throughout the Subsections 3.2 to 3.4.

Our main results are proved in Sections 4 and 5. In Section 4 we prove general
properties about the boundary representations constructed in Section 3 and
give specific description of them in particular cases. The Section 5 focuses
on spectral estimates.

Theorems 1 and 2 are respectively established in the self-contained Subsection
5.2. Theorems 3 and 4 are respectively consequences of Corollary 5.22 and 5.23
in Section 5.4. Subsection 5.3 is dedicated to the proof of Theorem 5. Theorem
6 is a consequence of the construction of Section 3, Subsection 4.1 and
Corollary 5.24. Theorem 7 together with Corollary 1.13 and 1.14 are proved in
subsection 4.2. The Proposition 1.11 is established in Subsection 4.1.

###  1.5. Notations and symbols

Given two real functions $a,b:S\rightarrow{\mathbb{R}}$ on a set $S$, we
denote $a\prec_{*}b$ if there exists a constant $C$ that depends on $*$ such
that $a(s)\leq Cb(s)$ for all $s\in S$ and $a\asymp_{*}b$ if $a\prec_{*}b$ and
$b\prec_{*}a$.
Given two real functions $a,b:S\rightarrow{\mathbb{R}}$ on a set $S$, we
denote $a\lesssim_{*}b$ if there exists a constant $C$ that depends on $*$
such that $a(s)\leq b(s)+C$ for all $s\in S$ and $a\simeq_{*}b$ if
$a\lesssim_{*}b$ and $b\lesssim_{*}a$.

###  1.6. Acknowledgments

The authors would like to express his gratitude to Uri Bader, Alex Furman and
Lam Pham for inspiring discussions that led to the present work as well as
Bogdan Nica and Jan Spakula for their support and feedbacks and Rhiannon
Dougall for interesting discussions about boundary representations.

The author was supported by EPSRC Standard Grant EP/V002899/1.

##  2\. Preliminaries

In this section we recall basics about Gromov hyperbolic geometry and counting
arguments. The reader can refer to [42] for more about Gromov hyperbolic
geometry .

Let $(X,d_{X})$ be a discrete uniformly locally finite metric space and $o\in
X$ a fixed basepoint.

###  2.1. Strong hyperbolicity and consequences

A metric space $(X,d_{X})$ is roughly geodesic if for all $x,y\in X$, there
exist a interval $I\subset{\mathbb{R}}$ and $c:I\rightarrow X$ such that
$d_{X}(c(t_{1}),c(t_{2}))\simeq_{X}|t_{1}-t_{2}|$ for all $t_{1},t_{2}\in I$.

The Gromov product on $(X,d_{X})$ at $o\in X$ is defined as

$(x,y)_{o}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{2}(d_{X}(x,o)+d_{X}(y,o)-d_{X}(x,y))$

for all $x,y\in X$ and $(X,d_{X})$ is Gromov hyperbolic if there exists $C\geq
0$ such that

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

for all $x,y,z\in X$.

Assuming $(X,d_{X})$ is Gromov hyperbolic the Gromov boundary of $(X,d_{X})$
can be defined as

${\partial}X\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(x_{n})_{n}\in
X^{\mathbb{N}}:\liminf_{n,m}(x_{n},x_{m})_{o}=+\infty\\}/\sim$

with
$(x_{n})_{n}\sim(y_{n})_{n}\Leftrightarrow\liminf_{n,m}(x_{n},y_{m})_{o}=+\infty$.
The space ${\partial}X$ endowed with the natural visual topology is compact as
the space $(X,d_{X})$ is proper. Moreover there exists a compatible topology
on $\overline{X}=X\cup{\partial}X$ that makes it a compact space [42].

######  Definition 2.1 ([68]).

A roughly geodesic hyperbolic metric space $(X,d_{X})$ is strongly hyperbolic
if there exists $\varepsilon>0$ such that

$e^{-\varepsilon(x,y)_{o}}\leq
e^{-\varepsilon(x,z)_{o}}+e^{-\varepsilon(z,y)_{o}}$

for all $x,y,z,o\in X$.

Assuming $(X,d_{X})$ is strongly hyperbolic, the Gromov product extends
continuously to the Gromov compactification $\overline{X}=X\cup{\partial}X$
(excluding the diagonal pairs in ${\partial}X$, i.e $(\xi,\xi)$ with
$\xi\in{\partial}X$) and a visual distance $d_{{\partial}X}$ on ${\partial}X$
can be defined by
$d_{{\partial}X}(\xi,\eta)=e^{-\varepsilon_{0}(\xi,\eta)_{o}}$ for all
$\xi,\eta\in{\partial}X$ where $\varepsilon_{0}>0$ is chosen small enough. In
particular
$\lim_{z\rightarrow\xi}d_{X}(x,z)-d_{X}(y,z)=b_{\xi}(x,y)=d_{X}(x,y)-2(x,\xi)_{y}$
for all $\xi\in{\partial}X$ and $x,y\in X$.

######  Theorem (Mineyev-Yu [65] ).

Every Gromov hyperbolic group admits an invariant roughly geodesic and
strongly hyperbolic distance on themselves that is quasi-isometric to a word
distance.

See [64] and [68] for more about the subject.

###  2.2. Orbital distribution

Given $x\in X$ we denote $\widehat{x}_{o}\subset{\partial}X$ the non-empty
compact subset defined as

$\widehat{x}_{o}=\\{\xi\in{\partial}X:(x,\xi)_{o}=\|(x,\bullet)_{o}\|_{\infty}\\}$

As a consequence of strong hyperbolicity the map $\xi\mapsto(\xi,x)_{o}$ for
$x\in X$ is continuous on the compact space ${\partial}X$.

######  Remark 2.2.

The hat map, $x\mapsto\widehat{x}_{o}$, can be formalised using the notion of
correspondences [3, Chap. 17].

Using a tree approximation argument one can show that

$(x,\xi)_{o}\simeq d_{X}(x,o)$

for all $x\in X$ and $\xi\in\widehat{x}_{o}$. Moreover, using strong
hyperbolicity, $d_{{\partial}X}(\xi,\eta)=e^{-\varepsilon(\xi,\eta)_{o}}\prec
e^{-\varepsilon d_{X}(x,o)}$ for all $\xi,\eta\in\widehat{x}_{o}$, in other
words $\text{diam}(\widehat{x}_{o})\prec e^{-\varepsilon d_{X}(x,o)}$. In the
rest we use the notation $\widehat{x}$ for any $\xi\in\widehat{x}_{o}$ and
relation that does not involve a particular choice.

There exist $C_{1},C_{2}>0$ and $L>R>0$ such that

$B(\widehat{x},C_{1}e^{-\varepsilon_{0}d_{X}(x,o)})\subset\mathcal{O}_{o}(x;R)\subset
B(\widehat{x},C_{2}e^{-\varepsilon_{0}d_{X}(x,o)})$

where
$\mathcal{O}_{o}(x;R)\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\xi\in{\partial}X:(\xi,x)_{o}\geq
d(x,o)-R\\}$ stands for $R$-shadow at $x\in X$ with $d_{X}(o,x)\geq L$ (see
[14, Prop. 2.1]). A sufficient condition for the above $R>0$ is that any rough
geodesic from $o$ to $\xi\in{\partial}X$ intersects

$C_{o,m}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in X:Rm\leq
d_{X}(x,o)<R(m+1)\\}$

and ${\partial}X=\bigcup_{x\in C_{o,m}}\mathcal{O}_{o}(x;R)$ for all
$m\in{\mathbb{N}}$ . The reader can refer to [41, Sect. 4] for elements of
proof (see also [8] [21]).

Observe that $\mathcal{O}_{o}(x;R)\cap\mathcal{O}_{o}(y;R)\neq\emptyset$
implies that $d_{X}(x,y)\leq 4R$ and as $(X,d_{X})$ is uniformly locally
finite the covering $\mathcal{C}_{o,m}=\\{\mathcal{O}_{o}(x;R)\\}_{x\in
C_{o,m}}$ have uniformly finite multiplicity independently of $m$, i.e there
exists $M\geq 1$, for all $\xi\in{\partial}X$, $|\\{x\in
C_{o,m}:\xi\in\mathcal{O}(x;R)\\}|\leq M$.

######  Definition 2.3.

Given for $D>0$, a metric measure space $(Z,d,\nu)$ is called $D$-David-
Ahlfors regular (or $D$-regular) if $\nu(B(\xi,r))\asymp r^{D}$ for all
$r\in[0,\text{diam}(Z,d)]$.

Equivalently [33] given a ball of radius $0<r<\text{diam}(Z,d)$, any maximal
$\varepsilon$-discrete sequence in $B(\xi,r)$ has a number of elements
equivalent, up to multiplicative constants independent of $\xi\in Z$ and $r$,
to $\frac{r^{D}}{\varepsilon^{D}}$.

We conclude this subsection with Vitali’s Lemma [47] and some consequences:

######  Lemma 2.4.

[Vitali] Given an arbitrary collection, $\mathcal{C}$, of balls in a separable
metric space $(Z,d)$ of uniformly bounded radius. There exists a countable
subcollection $\mathcal{C}^{\prime}$ of pairwise disjoint balls such that

$\bigcup_{\mathcal{C}}B=\bigcup_{\mathcal{C}^{\prime}}5B^{\prime}$

Since the cover $(\mathcal{O}_{o}(x;R))_{x\in C_{o,m}}$ has uniformly finite
multiplicity, the Vitali’s covering Lemma 2.4 implies:

######  Corollary 2.5.

If $({\partial}X,d_{{\partial}X})$ is $D$-regular with respect to his
Hausdorff measure (or any other measure [47]), then for all
$\xi\in{\partial}X$ and $0<r<\text{diam}({\partial}X,d_{{\partial}X})$ the
number of elements in $\mathcal{C}_{o,m}$ that intersect $B(\xi,r)$ is
equivalent, up to multiplicative constants independent of $\xi\in Z$ and $r$,
to $e^{\varepsilon_{0}DRm}r^{D}$.

######  Remark 2.6.

In the case $(X,d_{X})=(\Gamma,d_{\Gamma})$ considered throughout the rest of
this work, one has
$\varepsilon_{0}\text{Dim}_{\text{Haus}}({\partial}\Gamma,d_{{\partial}\Gamma})=\delta_{\Gamma}$,
where $\delta_{\Gamma}$ is the critical exponent of $(\Gamma,d_{\Gamma})$
[29].

######  Definition 2.7.

A discrete metric space $(Y,d_{Y})$ is called purely spherical at $y\in Y$ if
there exists $R$, $|B_{Y}(y,r)|\asymp e^{\delta_{Y}r}$ for $r\geq R$ where
$\delta_{Y}$ for the critical exponent of $(Y,d_{Y})$.

######  Lemma 2.8.

The space $(X,d_{X})$ is purely spherical at $o\in X$ whenever ${\partial}X$
is David-Ahlfors regular.

###### Proof.

Let $C_{o,m}^{*}\subset C_{o,m}$ maximal such that ${\partial}X=\bigcup_{x\in
C_{o,m}^{*}}B(\widehat{x},5C_{2}e^{-\varepsilon_{0}d_{X}(x,o)})$ and
$(B(\widehat{x},C_{2}e^{-\varepsilon_{0}d_{X}(x,o)}))_{x\in C_{o,m}^{*}}$
pairwise disjoint. Then using $D$-regularity
$|C_{o,m}^{*}|e^{-\varepsilon_{0}DRm}\asymp 1$.

On the other hand it follows from maximality that for all $x\in C_{o,m}$,
there exists $x^{\prime}\in C_{o,m}^{*}$ such that
$B(\widehat{x},C_{2}e^{-\varepsilon_{0}d_{X}(x,o)})\cap
B(\widehat{x}^{\prime},C_{2}e^{-\varepsilon_{0}d_{X}(x,o)})\neq\emptyset$ and
uniform multiplicity that the number of $x$ that intersect a given ball
$B(\widehat{x}^{\prime},C_{2}e^{-\varepsilon_{0}d_{X}(x,o)})$ is uniformly
controlled. In particular $|C_{o,m}^{*}|\asymp|C_{o,m}|$. ∎

Let

$\mathcal{O}_{o}(m,n;y)\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in
C_{o,m}:Rn\leq(x,y)_{o}<R(n+1)\\}$
$\mathcal{O}_{o}^{\prime}(m,n;y)\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in
C_{o,m}:\exists\xi\in\widehat{x},Rn\leq(\xi,y)_{o}<R(n+1)\\}$

for $y\in X\cup{\partial}X$, $m,n\in{\mathbb{N}}$ with $n\leq m$ and $C\geq
e^{\varepsilon_{0}R}$. It follows from Corollary 2.5 that

(1)
$\frac{|\mathcal{O}_{o}^{\prime}(m,n;y)|}{|C_{o,m}|}\asymp\frac{|\mathcal{O}_{o}(m,n;y)|}{|C_{o,m}|}\asymp
e^{-\delta Rn}$

whenever ${\partial}X$ is $D$-regular (see also [41, Lem. 4.3] for alternative
proof).

###  2.3. $s$-density representations and summation estimates

In the rest $(X,d_{X})=(\Gamma,d_{\Gamma})$, $o=e$ and we denote
$C_{m}=C_{o,m}$. In particular ${\partial}\Gamma$ is $D$-regular with respect
to his visual distance due to the Patterson-Sullivan theory and
$\varepsilon_{0}D_{(\varepsilon_{0})}=\delta$ (see [29, Thm. 7.7]).

######  Definition 2.9.

A slow growing function is a non-decreasing function
$\theta:{\mathbb{R}}_{+}\rightarrow[1,+\infty)$ such that for all
$\varepsilon>0$, there exists $T$, for all $u\geq 0$ and $t\geq T$:
$\theta(u+t)\leq e^{\varepsilon u}\theta(t)$.

Note that $\theta=1$ is a slow growing function.

Given $\theta$ and $s\in[0,1]$, we define the $\theta$-rearranged $s$-density
representation, $\pi_{\theta,s}$, on ${\mathcal{C}}(\overline{\Gamma})$ as

$\pi_{\theta,s}(g)\phi(x)=e^{-s\delta
b_{x}(g)}\frac{\theta(d_{\Gamma}(x,g))}{\theta(d_{\Gamma}(x,e))}\phi(g^{-1}x)$

for all $x\in\overline{\Gamma}$. Observe that the continuous extension over
the boundary, ${\partial}\Gamma$, satisfies
$\pi_{\theta,s}|_{{\partial}\Gamma}=\pi_{s}|_{{\partial}\Gamma}=\pi_{1,s}|_{{\partial}\Gamma}$.

The following estimates are extensively used in the rest:

######  Lemma 2.10 (Version 1).

Let $\sigma_{2}\in(\frac{1}{2},1]$, $\sigma_{1}\in{\mathbb{R}}_{+}$,
$\pi_{\sigma_{2}}$ the $\sigma_{2}$-density representation and

$\mathcal{S}_{m}(\sigma_{1},\sigma_{2};h)\stackrel{{\scriptstyle\text{def}}}{{=}}(\sum_{g\in
C_{m}}e^{-\sigma_{1}\delta d_{\Gamma}(g,e)}\pi_{\sigma_{2}}(g)){\bf 1}_{h}$

Then

$\mathcal{S}_{m}(\sigma_{1},\sigma_{2};h)\asymp_{\sigma_{2}}\begin{cases}e^{(1-(\sigma_{1}+\sigma_{2}))\delta
Rm}e^{(2\sigma_{2}-1)\delta\min\\{Rm,d_{\Gamma}(h,e)\\}}&\text{ for
$h\in\Gamma$}\\\ e^{(\sigma_{2}-\sigma_{1})\delta
Rm}&\text{otherwise}\end{cases}$

###### Proof.

For $h\in\Gamma$:

$\displaystyle\mathcal{S}_{m}(\sigma_{1},\sigma_{2};h)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\sum_{g\in
C_{m}}e^{-\sigma_{1}\delta d_{\Gamma}(g,e)}\pi_{\sigma_{2}}(g)){\bf 1}_{h}$
$\displaystyle=\sum_{g\in C_{m}}e^{-\sigma_{1}\delta
d_{\Gamma}(g,e)}e^{-\sigma_{2}\delta b_{h}(g^{-1},e)}$
$\displaystyle=\sum_{g\in
C_{m}}e^{2\sigma_{2}\delta(g,h)_{e}}e^{-(\sigma_{1}+\sigma_{2})\delta
d_{\Gamma}(g,e)}$ $\displaystyle\asymp e^{(1-(\sigma_{1}+\sigma_{2}))\delta
Rm}[\frac{1}{|C_{m}|}\sum_{C_{m}}e^{2\sigma_{2}\delta(g,h)_{e}}]$

Using the estimate (1):

$\displaystyle\frac{1}{|C_{m}|}\sum_{C_{m}}e^{2\sigma_{2}\delta(g,h)_{e}}\asymp\sum_{n:Rn\leq\min\\{Rm,d_{\Gamma}(h,e)\\}}\frac{|\mathcal{O}(m,n;h)|}{|C_{m}|}e^{2\sigma_{2}\delta
Rn}$
$\displaystyle\asymp\sum_{n:Rn\leq\min\\{Rm,d_{\Gamma}(h,e)\\}}e^{(2\sigma_{2}-1)\delta
Rn}\asymp_{\sigma_{2}}e^{(2\sigma_{2}-1)\delta\min\\{Rm,d_{\Gamma}(h,e)\\}}$

and thus

$\displaystyle\mathcal{S}_{m}(\sigma_{1},\sigma_{2};h)$
$\displaystyle\asymp_{\sigma_{2}}e^{(1-(\sigma_{1}+\sigma_{2}))\delta
Rm}e^{(2\sigma_{2}-1)\delta\min\\{Rm,d_{\Gamma}(h,e)\\}}$

The case $h=\xi\in{\partial}\Gamma$ follows from similar computations
replacing $\min\\{Rm,d_{\Gamma}(h,e)\\}$ with $Rm$.

∎

######  Lemma 2.11 (Version 2).

Let $\theta$ be a slow growing function, $\sigma_{2}\in(\frac{1}{2},1]$,
$\sigma_{1}\in{\mathbb{R}}_{+}$, $\pi_{\theta,\sigma_{2}}$ the rearranged
$\sigma_{2}$-density representation and

$\mathcal{S}_{\theta,m}(\sigma_{1},\sigma_{2};h)\stackrel{{\scriptstyle\text{def}}}{{=}}(\sum_{g\in
C_{m}}\theta(d_{\Gamma}(g,e))e^{-\sigma_{1}\delta
d_{\Gamma}(g,e)}\pi_{\sigma_{2},\theta}(g)){\bf 1}_{h}$

for $\sigma_{2}>\frac{1}{2}$. Then for all $\varepsilon>0$, there exists
$N\geq 0$ such that:

$\mathcal{S}_{\theta,m}(\sigma_{1},\sigma_{2};h)\prec_{\sigma_{2}}\begin{cases}\theta^{2}(Rm)e^{\varepsilon
d_{\Gamma}(h,e)}e^{(1-(\sigma_{1}+\sigma_{2}))\delta
Rm}e^{(2\sigma_{2}-1)\delta\min\\{Rm,d_{\Gamma}(h,e)\\}}&\text{ for
$h\in\Gamma$}\\\ \theta^{2}(Rm)e^{\varepsilon
d_{\Gamma}(h,e)}e^{(\sigma_{2}-\sigma_{1})\delta
Rm}&\text{otherwise}\end{cases}$

for all $m\geq N$.

###### Proof.

First observe that if $\theta$ has slow growth then $\theta^{2}$ as well and
$\theta\leq\theta^{2}$ as $\theta\geq 1$. Let

$k_{\theta}(g,h)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{\theta(d_{\Gamma}(h,g))}{\theta(d_{\Gamma}(h,e))\theta(d_{\Gamma}(g,e))}\leq\frac{\theta(d_{\Gamma}(g,e)+d_{\Gamma}(h,e))}{\theta(d_{\Gamma}(h,e))\theta(d_{\Gamma}(g,e))}$

for $g,h\in\Gamma$. Given $\varepsilon>0$ and $T_{\varepsilon}$ as in
Definition 2.9, for $d_{\Gamma}(g,e)\geq T_{\varepsilon}$ one has:

$k_{\theta}(g,h)\leq\frac{e^{\varepsilon
d_{\Gamma}(h,e)}}{\theta(d_{\Gamma}(h,e))}$

for all $h\in\Gamma$.

Given $h\in\Gamma$:

$\displaystyle\mathcal{S}_{\theta,m}(\sigma_{1},\sigma_{2};h)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\sum_{g\in
C_{m}}\theta(d_{\Gamma}(g,e))e^{-\sigma_{1}\delta
d_{\Gamma}(g,e)}\pi_{\theta,\sigma_{2}}(g)){\bf 1}_{h}$
$\displaystyle=\sum_{g\in C_{m}}\theta(d_{\Gamma}(g,e))e^{-\sigma_{1}\delta
d_{\Gamma}(g,e)}\frac{\theta(d_{\Gamma}(h,g))}{\theta(d_{\Gamma}(h,e))}e^{-\sigma_{2}\delta
b_{h}(g^{-1},e)}$ $\displaystyle=\sum_{g\in
C_{m}}k_{\theta}(g,h)e^{2\sigma_{2}\delta(g,h)_{e}}\theta^{2}(d_{\Gamma}(g,e))e^{-(\sigma_{1}+\sigma_{2})\delta
d_{\Gamma}(g,e)}$
$\displaystyle\asymp\theta^{2}(Rm)e^{(1-(\sigma_{1}+\sigma_{2}))\delta
Rm}[\frac{1}{|C_{m}|}\sum_{g\in
C_{m}}k_{\theta}(g,h)e^{2\sigma_{2}\delta(g,h)_{e}}]$

For $m\geq T_{\varepsilon}$, doing a similar computation as in Lemma 2.10
proof:

$\displaystyle\frac{1}{|C_{m}|}\sum_{g\in
C_{m}}k_{\theta}(g,h)e^{2\sigma_{2}(g,h)_{e}}\prec\frac{e^{\varepsilon
d_{\Gamma}(h,e)}}{\theta(d_{\Gamma}(h,e))}\frac{1}{|C_{m}|}\sum_{g\in
C_{m}}e^{2\sigma_{2}(g,h)_{e}}$ $\displaystyle\prec_{\sigma_{2}}e^{\varepsilon
d_{\Gamma}(h,e)}e^{(2\sigma_{2}-1)\delta\min\\{Rm,d_{\Gamma}(h,e)\\}}$

and thus:

$\displaystyle\mathcal{S}_{\theta,m}(\sigma_{1},\sigma_{2};h)$
$\displaystyle\prec_{\sigma_{2}}\theta^{2}(Rm)e^{\varepsilon
d_{\Gamma}(h,e)}e^{(1-(\sigma_{1}+\sigma_{2}))\delta
Rm}e^{(2\sigma_{2}-1)\delta\min\\{Rm,d_{\Gamma}(h,e)\\}}$

The case $h=\xi\in{\partial}\Gamma$ follows from similar computations
replacing $\min\\{Rm,d_{\Gamma}(h,e)\\}$ with $Rm$ and using the fact that
$\frac{\theta(d_{\Gamma}(h,g))}{\theta(d_{\Gamma}(h,e))}\xrightarrow{d_{\Gamma}(h,e)\rightarrow\infty}1$.

∎

######  Remark 2.12.

Following the same lines as Lemma 2.10:
Given

$\mathcal{S}_{m}(\sigma,\frac{1}{2};h)\stackrel{{\scriptstyle\text{def}}}{{=}}(\sum_{g\in
C_{m}}e^{-\sigma\delta d_{\Gamma}(g,e)}\pi_{\frac{1}{2}}(g)){\bf 1}_{h}$

with $\sigma\in{\mathbb{R}}_{+}$, there exist $N>0$ and $L>0$ such that

$\mathcal{S}_{m}(\sigma,\frac{1}{2};h)\asymp\begin{cases}e^{(\frac{1}{2}-\sigma)\delta
Rm}\min\\{Rm,d_{\Gamma}(h,e)\\}&\text{ for $h\in\Gamma$}\\\
e^{(\frac{1}{2}-\sigma)\delta Rm}Rm&\text{otherwise}\end{cases}$

for $m\geq N$, $d_{\Gamma}(h,e)\geq L$.

######  Corollary 2.13.

Let $\sigma_{2}\in[\frac{1}{2},1]$, $\sigma_{1}\in{\mathbb{R}}_{+}$ with
$\sigma_{1}>\sigma_{2}$, $\pi_{\sigma_{2}}$ the $\sigma_{2}$-density
representation and

$\mathcal{S}_{\infty}(\sigma_{1},\sigma_{2};\xi)=(\sum_{g\in\Gamma}e^{-\sigma_{1}\delta
d_{\Gamma}(g,e)}\pi_{\sigma_{2}}(g)){\bf 1}_{\xi}$

for $\xi\in{\partial}\Gamma$. Then

$\mathcal{S}_{\infty}(\sigma_{1},\sigma_{2};\xi)\asymp_{\sigma_{2}}\begin{cases}\frac{1}{\sigma_{1}-\sigma_{2}}&\text{for
$\sigma_{2}>\frac{1}{2}$}\\\ \frac{1}{(\sigma_{1}-\frac{1}{2})^{2}}&\text{for
$\sigma_{2}=\frac{1}{2}$}\end{cases}$

In particular $\mathcal{S}_{\infty}(\sigma_{1},\sigma_{2};\bullet)$ is bounded
on ${\partial}\Gamma$.

###### Proof.

Given $\xi\in{\partial}\Gamma$ and $\sigma_{2}>\frac{1}{2}$, one has:

$\displaystyle\mathcal{S}_{\infty}(\sigma_{1},\sigma_{2};\xi)$
$\displaystyle=(\sum_{g\in\Gamma}e^{-\sigma_{1}\delta
d_{\Gamma}(g,e)}\pi_{\sigma_{2}}(g)){\bf
1}_{\xi}=\sum_{m}\mathcal{S}_{m}(\sigma_{1},\sigma_{2};\xi)$
$\displaystyle\asymp_{\text{Lem.
\ref{lem:tech:v1}}}\sum_{m}e^{(\sigma_{2}-\sigma_{1})\delta Rm}$
$\displaystyle\asymp\frac{1}{\sigma_{1}-\sigma_{2}}$

The case $\sigma_{2}=\frac{1}{2}$ follows from similar computations using
remark 2.12 and a summation by part argument. ∎

######  Corollary 2.14.

Let $s\in[\frac{1}{2},1]$ and $\pi_{s}$ the $s$-density representation. Then
there exists $L_{0}$, for all $L\geq L_{0}$:

$\sum_{g:d_{\Gamma}(g,e)\leq L}\pi_{s}(g){\bf
1}|_{{\partial}\Gamma}\asymp_{s}\begin{cases}e^{s\delta L}&\text{for
$s>\frac{1}{2}$}\\\ e^{s\delta L}L&\text{for $s=\frac{1}{2}$}\end{cases}$

Moreover

$\sum_{g:d_{\Gamma}(g,e)\leq L}\pi_{s}(g){\bf
1}|_{\overline{\Gamma}}\prec_{s}\begin{cases}e^{s\delta L}&\text{for
$s>\frac{1}{2}$}\\\ e^{s\delta L}L&\text{for $s=\frac{1}{2}$}\end{cases}$

###### Proof.

Given $\xi\in{\partial}\Gamma$ and $\sigma_{2}>\frac{1}{2}$, one has:

$\displaystyle\sum_{g:d_{\Gamma}(g,e)\leq L}\pi_{s}(g){\bf 1}_{\xi}$
$\displaystyle=\sum_{m}\mathcal{S}_{m}(0,s;\xi)\asymp_{s,\text{Lem.
\ref{lem:tech:v1}}}\sum_{m:mR\leq L}e^{s\delta Rm}$
$\displaystyle\asymp_{s}e^{s\delta L}$

The case $\sigma_{2}=\frac{1}{2}$ follows from similar computations using
remark 2.12 and summation by part. The upper bound for
$x\in\Gamma\subset\overline{\Gamma}$ follows from the fact that
$\min\\{Rm,d_{\Gamma}(h,e)\\}\leq Rm$ and the above computations. ∎

###  2.4. Nets and positive families of vectors

The reader can refer to [3, Chap. 2] for details about the notion of net and
order.

Given a Hilbert space, $\mathcal{H}$, and countable set, $I$, a family of
vectors $\\{v_{i}\\}_{i\in I}$ belongs to a positive cone in $\mathcal{H}$ if
$(v_{i},v_{j})\geq 0$ for all $i,j\in I$.

Given such a family, $\\{v_{i}\\}_{i\in I}$, we define the summation net as

$(2^{I}_{0},\subset)\rightarrow(\mathcal{H},\mathcal{H}_{+});F\subset I\mapsto
v_{F}=\sum_{F}v_{i}$

where $(2^{I}_{0},\subset)$ stands for the finite subsets of $I$ with direct
order induced by the inclusion. Observe that $(\|v_{F}\|)_{F}$ is a non-
decreasing net as

$\|v+w\|^{2}=\|v\|^{2}+\|w\|^{2}+2(v,w)\geq\max\\{\|v\|^{2},\|w\|^{2}\\}$

for all $v,w\in\mathcal{C}_{+}$ with
$\mathcal{C}_{+}\stackrel{{\scriptstyle\text{def}}}{{=}}\overline{\\{\sum_{F}c_{i}v_{i}:\text{$c_{i}\geq
0$ for all $i\in F$}\\}}\subset\mathcal{H}$.

######  Lemma 2.15.

Given a family $\\{v_{i}\\}_{i\in I}$ within a positive cone of $\mathcal{H}$,
the summation net $(v_{F})_{F}$ converges if and only if $(\|v_{F}\|)_{F}$ is
bounded.

In particular $\sum_{i\in I}v_{i}$ is well defined iff there exists an
exhaustion $(F_{n})_{n}$ by finite sets of $I$ such that
$\sup_{n}\|\sum_{F_{n}}v_{i}\|$ is finite.

###### Proof.

It is enough to prove that if $(\|v_{F}\|)_{F}$ is bounded the net
$(v_{F})_{F}$ converges in $\mathcal{H}$.

As $(\|v_{F}\|)_{F}$ is monotone and bounded it converges to
$\sup_{2^{I}_{0}}\|v_{F}\|$. In particular $(\|v_{F}\|)_{F}$ is a Cauchy net.
On the other hand given $F_{0}\subset F$ one has:

$\|v_{F\setminus
F_{0}}\|^{2}=\|v_{F}-v_{F_{0}}\|^{2}=\|v_{F}\|^{2}-\|v_{F_{0}}\|^{2}-2(v_{F_{0}},v_{F\setminus
F_{0}})\leq\|v_{F}\|^{2}-\|v_{F_{0}}\|^{2}$

that is small whenever $F_{0}$ is taken large enough and as

$\|v_{F}-v_{F^{\prime}}\|\leq\|v_{F^{\prime}\setminus F_{0}}\|+\|v_{F\setminus
F_{0}}\|$

for $F_{0}\subset F^{\prime},F$ it follows that $(v_{F})_{F}$ is a Cauchy net
in the Banach space $\mathcal{H}$ and thus converges to a unique limit point.
∎

######  Lemma 2.16.

Given a summable family $\\{v_{i}\\}_{i\in I}$ within a positive cone of
$\mathcal{H}$ and $u=(u_{i})_{i\in I},u^{\prime}=(u^{\prime}_{i})_{i\in
I}\in\ell^{\infty}_{+}(I)$ with $u\leq u^{\prime}$. Then

$\|\sum_{i\in I}u_{i}v_{i}\|\leq\|\sum_{i\in
I}u_{i}^{\prime}v_{i}\|\leq\|u^{\prime}\|_{\infty}\|\sum_{i\in I}v_{i}\|$

###### Proof.

It enough to prove the first inequality on a finite subset, $F$, of $I$. The
result will follow from Lemma 2.15.

In this case

$\displaystyle\|\sum_{i\in F}u_{i}v_{i}\|^{2}$ $\displaystyle=\sum_{i,j\in
F}u_{i}u_{j}(v_{i},v_{j})\leq\sum_{i,j\in
F}u_{i}^{\prime}u_{j}^{\prime}(v_{i},v_{j})=\|\sum_{i\in
F}u_{i}^{\prime}v_{i}\|^{2}$

∎

###  2.5. Observation about the GNS of pointwise positive functions

A function $\phi$ is called positive definite if the quadratic form on
${\mathbb{C}}[\Gamma]$ given by

$B_{\phi}(f)=\sum_{g,h}f(g)\overline{f(h)}\phi(g^{-1}h)$

is positive [10, Sect. 1.B]. As it is invariant by left translation it defines
a pre-unitary cyclic representation with $\text{Dir}_{e}$ as cyclic vector.
Moreover one has $\phi(g)=B_{\phi}(\lambda(g)\text{Dir}_{e},\text{Dir}_{e})$
for all $g\in\Gamma$. This construction, called Gelfand-Naimak-Segal
construction (or GNS), defines a bijection between positive definite functions
and classes of cyclic representations of $\Gamma$.

Let $\mathbb{P}_{+}(\Gamma)$ be the set of positive definite functions on
$\Gamma$ that are pointwise positive. Given $\phi\in\mathbb{P}_{+}(\Gamma)$ we
denote $(\pi_{\phi},\mathcal{H}_{\phi},{\bf 1}_{\phi})$ its GNS triple.

Observation that $(\pi_{\phi}(g){\bf 1}_{\phi})_{g}$ belongs to a positive
cones of $\mathcal{H}_{\phi}$ as defined in Subsection 2.4. This follows from
the fact that

$(\pi_{\phi}(g){\bf 1}_{\phi},\pi_{\phi}(h){\bf
1}_{\phi})=(\pi_{\phi}(h^{-1}g){\bf 1}_{\phi},{\bf
1}_{\phi})=\phi(h^{-1}g)\geq 0$

Given a weight $\omega$ with $\omega(g)>0$ for all $g\in\Gamma$ such that
$\sum_{g}\omega(g)\pi_{\phi}(g){\bf 1}_{\phi}$ is well defined, Lemma 2.16
implies that $\sum_{g}\omega(g)|f|(g)\pi_{\phi}(g){\bf 1}_{\phi}$ is also well
defined with

$\|\sum_{g}\omega(g)|f|(g)\pi_{\phi}(g){\bf
1}_{\phi}\|\leq\|f\|_{\infty}\|\sum_{g}\omega(g)\pi_{\phi}(g){\bf 1}_{\phi}\|$

for all $f\in\ell^{\infty}(\Gamma)$. One can construct a pre-Hilbertian
structure, $B_{\phi,\omega}$, on $\ell^{\infty}(\Gamma)$ as follows:

$\displaystyle\mathcal{P}_{\omega}^{2}.B_{\phi,\omega}(f)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{g,h\in\Gamma}f(g)f(h)\phi(g^{-1}h)\omega(g)\omega(h)$
$\displaystyle=_{\text{Lem.
\ref{lem:sum:net}}}\|\sum_{g}\omega(g)f(g)\pi_{\phi}(g){\bf 1}_{\phi}\|^{2}$
$\displaystyle=\|\sum_{g}\omega(g)f_{+}(g)\pi_{\phi}(g){\bf
1}_{\phi}-\sum_{g}\omega(g)f_{-}(g)\pi_{\phi}(g){\bf 1}_{\phi}\|^{2}$
$\displaystyle\leq 2\|\sum_{g}\omega(g)f_{+}(g)\pi_{\phi}(g){\bf
1}_{\phi}\|^{2}+2\|\sum_{g}\omega(g)f_{-}(g)\pi_{\phi}(g){\bf 1}_{\phi}\|^{2}$
$\displaystyle\leq_{\text{Lem.
\ref{lem:infty:net}}}4\|f\|_{\infty}^{2}\|\sum_{g}\omega(g)\pi_{\phi}(g){\bf
1}_{\phi}\|^{2}$

with $\|\sum_{g}\omega(g)\pi_{\phi}(g){\bf
1}_{\phi}\|^{2}=\mathcal{P}_{\omega}^{2}$ and $f\in\ell^{\infty}(\Gamma)$. The
case of complex-valued functions is obtained by scalar extension.

For this positive quadratic form the linear operators

$\pi_{\omega}(g)f(z)=\frac{\omega(g^{-1}z)}{\omega(z)}f(g^{-1}z)$

for $g,z\in\Gamma$ defines a pre-unitary representation.

Observe that the operator

$J_{\omega}:\ell^{\infty}(\Gamma)\rightarrow\mathcal{H}_{\phi};\quad
f\mapsto\frac{1}{\mathcal{P}_{\omega}}\sum_{z}\omega(z)f(z)\pi_{\phi}(z){\bf
1}_{\phi}$

defines a bounded $(\pi_{\omega},\pi)$-intertwiner with dense range. Indeed
given $f^{\prime}\in{\mathbb{C}}[\Gamma]$ one has $\pi_{\phi}(f^{\prime}){\bf
1}=J_{\omega}(f)$ with $f(g)=\mathcal{P}_{\omega}\omega(g)^{-1}f^{\prime}(g)$.

The operator $J_{\omega}$ naturally extends to a Hilbert space of functions:

Observe that

$\displaystyle
B_{\phi,\omega}(f_{1},f_{2})=\mathcal{P}_{\omega}^{-2}\sum_{x,y\in\Gamma}f_{1}(x)f_{2}(y)\omega(x)\omega(y)\phi(x^{-1}y)$
$\displaystyle=\sum_{x\in\Gamma}f_{1}(x)\frac{\sum_{y\in\Gamma}f_{2}(y)\omega(y)\phi(x^{-1}y)}{\sum_{y\in\Gamma}\omega(y)\phi(x^{-1}y)}\mathcal{P}_{\omega}^{-2}.\omega(x)\sum_{y\in\Gamma}\omega(y)\phi(x^{-1}y)$
$\displaystyle=\int_{\Gamma}f_{1}(x)E_{\omega}(f_{2})_{x}d\mu_{\omega}(x)$

where $\mu_{\omega}$ denote a probability on $\overline{\Gamma}$ defined as:

$\int_{\Gamma}f(x)d\mu_{\omega}(x)=\mathcal{P}_{\omega}^{-2}\int_{\Gamma}f(x)\omega(x)\sum_{y\in\Gamma}\omega(y)\phi(x^{-1}y)$

and $E_{\omega}$ the $(\overline{\Gamma},\mu_{\omega})$-conditional
expectation ([34, Chap. ii]):

$E_{\omega}(f)_{x}=\frac{\sum_{y\in\Gamma}f(y)\omega(y)\phi(x^{-1}y)}{\sum_{y\in\Gamma}\omega(y)\phi(x^{-1}y)}$

for $f\in L^{1}(\mu_{\omega})$.

In particular $E_{\omega}$ induces a self-adjoint contraction on
$L^{2}(\mu_{\omega})$ with $E_{\omega}({\bf 1})={\bf 1}$ and thus
$B_{\phi,\omega}(f)=\|J_{\omega}(f)|\mathcal{H}_{\phi}\|^{2}\leq\|f|L^{2}(\mu_{\omega})\|^{2}$
for $f\in\ell^{\infty}(\Gamma)$. It follows that $J_{\omega}$ extends to
$L^{2}(\mu_{\omega})$.

Moreover

$E_{\omega}(\pi_{\omega}(g)f)_{x}=\frac{\sum_{y\in\Gamma}\frac{\omega(g^{-1}y)}{\omega(y)}\omega(y)\phi(x^{-1}y)}{\sum_{y\in\Gamma}\omega(y)\phi(x^{-1}y)}E_{\omega}(f)_{g^{-1}x}$

in other words $E_{\omega}(\pi_{\omega}(g)f)=\pi_{\omega}^{*}(g)E_{\omega}(f)$
where $f\in L^{2}(\mu_{\omega})$ and $\pi_{\omega}^{*}$ denote the adjoint of
$\pi_{\omega}$ in $L^{2}(\mu_{\omega})$. In particular $\text{Ker}J_{\omega}$
is $\pi_{\omega}$-invariant and $J_{\omega}|_{N}$ with
$N=L^{2}(\mu_{\omega})/{\text{Ker}J_{\omega}}\simeq\text{Ker}J_{\omega}^{\perp}$
extends as a unitary equivalence between the completion of $N$ with respect to
the quadratic form $B_{\phi,\omega}$, $\overline{(N,B_{\phi,\omega})}$, and
$(\pi_{\phi},\mathcal{H}_{\phi})$ and satisfies

$[J_{\omega}|_{\overline{N}}]^{-1}\pi_{\phi}J_{\omega}|_{\overline{N}}=\pi_{\omega}$

##  3\. Boundary representations of positive cyclic representations

This section is dedicated to the construction of boundary representations
associated to positive cyclic representations. Those representations are
obtained as limit of twisted GNS construction (see Subsection 2.5).

###  3.1. Critical exponent and Poincaré series

Given a pointwise positive function on $\Gamma$, $f$, the normalised critical
exponent of $f$ is defined as the infimum of $\sigma>0$ such that:

$\mathcal{P}(f;\sigma)=\sum_{g\in\Gamma}e^{-\sigma\delta d_{\Gamma}(g,e)}f(g)$

is finite and denoted $s(f)$. The critical exponent of $f$ is
$\delta(f)\stackrel{{\scriptstyle\text{def}}}{{=}}s(f)\delta$. A pointwise
positive function is of divergent type if
$\mathcal{P}(f;\sigma)\xrightarrow{\sigma\rightarrow s(f)^{+}}+\infty$.

######  Remark 3.1.

The indicator function of a subgroup of $\Gamma$ is of divergent type if and
only if the subgroup is of divergent type [71].

As in the previous constructions (e.g [29] [30] [71]) the divergence of the
Poincaré series $\mathcal{P}(f;\sigma)$ at $\delta(f)$ is necessary. This can
be arranged by an adaptation of the so-called Patterson argument [71]. The
following sequential formulation of Patterson’s trick is used later to deal
with families of positive cyclic representations.

######  Lemma 3.2.

Let $(f_{n})_{n}$ be a sequence of non-zero pointwise positive functions on
$\Gamma$ with $\lim_{n}\delta(f_{n})=\delta_{\infty}$ finite. Let
$(\sigma_{n})_{n}$ be a non-increasing sequence such that
$\sigma_{n}\delta>\delta(f_{n})$ and
$\sigma_{n}\delta\rightarrow\delta_{\infty}$. There exists a non-decreasing
map $\theta:{\mathbb{R}}_{+}\rightarrow[1,+\infty)$ such that:

  1. (1)

for all $\varepsilon>0$, there exists $T$, for all $u\geq 0$ and $t\geq T$:
$\theta(u+t)\leq e^{\varepsilon u}\theta(t)$, i.e $\theta$ has slow growth;

  2. (2)

$\mathcal{P}_{\theta}(f_{n};\sigma)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{g\in\Gamma}f_{n}(g)\theta(d_{\Gamma}(g,e))e^{-\sigma\delta
d_{\Gamma}(g,e)}$

is bounded when $\delta(f_{n})<\sigma\delta$ and unbounded when
$\delta(f_{n})>\sigma\delta$;

  3. (3)

$\lim_{n}\mathcal{P}_{\theta}(f_{n};\sigma_{n})\rightarrow+\infty$.

###### Proof.

Let $(\varepsilon_{n})_{n}$ be a non-increasing sequence such that
$\varepsilon_{n}>0$, $\lim_{n}\varepsilon_{n}=0$ and
$(\sigma_{n}-\varepsilon_{n})\delta<\delta(f_{n})$.

We construct the function $\theta$ together with a non-decreasing positive
sequence $(t_{n})_{n}$ inductively as follows. Set $t_{0}=0$ and
$\theta(t_{0})=1$ and define $t_{n+1}$ such that $t_{n+1}>t_{n}+1$ and

$e^{-\varepsilon_{n}t_{n}}\theta(t_{n})\sum_{\\{g:t_{n}<d_{\Gamma}(g,e)\leq
t_{n+1}\\}}e^{-(\sigma_{n}-\varepsilon_{n})\delta d_{\Gamma}(g,e)}f_{n}(g)\geq
n$

and $\theta$ on $]t_{n},t_{n+1}]$ to be
$\theta(t)=e^{\varepsilon_{n}(t-t_{n})}\theta(t_{n})$.

Observe that

$\displaystyle\mathcal{P}_{\theta}(f_{n};\sigma_{n})$
$\displaystyle\geq\sum_{\\{g:t_{n}<d_{\Gamma}(g,e)\leq
t_{n+1}\\}}\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}f_{n}(g)$
$\displaystyle=e^{-\varepsilon_{n}t_{n}}\theta(t_{n})\sum_{\\{g:t_{n}<d_{\Gamma}(g,e)\leq
t_{n+1}\\}}e^{-(\sigma_{n}-\varepsilon_{n})\delta d_{\Gamma}(g,e)}f_{n}(g)\geq
n$

The rest follows from the definition of $\theta$.

∎

######  Proposition 3.3.

Let $f\neq 0$ be a positive function on $\Gamma$ with $\delta(f)$ finite and
$\theta$ as in Lemma 3.5. Then

$\mathcal{P}_{\theta,2}(f;\sigma)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{g,h\in\Gamma}f(g^{-1}h)\theta(d_{\Gamma}(g,e))e^{-\sigma\delta
d_{\Gamma}(g,e)}\theta(d_{\Gamma}(h,e))e^{-\sigma\delta d_{\Gamma}(h,e)}$

is bounded for $\max\\{\frac{1}{2}\delta,\delta(f)\\}<\sigma\delta$, unbounded
for $\max\\{\frac{1}{2}\delta,\delta(f)\\}>\sigma\delta$ and

$\mathcal{P}_{\theta,2}(f;\sigma)\xrightarrow{\sigma\rightarrow\max\\{\frac{1}{2}\delta,\delta(f)\\}^{+}}+\infty$

###### Proof.

Let $\gamma\in\\{f>0\\}$ and $\sigma>0$ such that
$\mathcal{P}_{\theta,2}(f;\sigma)$ is finite. Then

$\displaystyle f(\gamma)$
$\displaystyle\sum_{g\in\Gamma}(\theta(d_{\Gamma}(g,e)))^{2}e^{-2\sigma\delta
d_{\Gamma}(g,e)}$
$\displaystyle\asymp_{\theta,\gamma}f(\gamma)\sum_{g\in\Gamma}\theta(d_{\Gamma}(g\gamma,e))e^{-\sigma\delta
d_{\Gamma}(g\gamma,e)}\theta(d_{\Gamma}(g,e))e^{-\sigma\delta
d_{\Gamma}(g,e)}$ $\displaystyle\leq\mathcal{P}_{\theta,2}(f;\sigma)$

and thus we must have $\sigma>\frac{1}{2}$. On the other hand

$\mathcal{P}_{\theta}(f;\sigma)=\sum_{\Gamma}e^{-\sigma\delta
d_{\Gamma}(g,e)}\theta(d_{\Gamma}(g,e))f(g)\leq\mathcal{P}_{\theta,2}(f;\sigma)$

and thus
$\mathcal{P}_{\theta,2}(f;\sigma)\xrightarrow{\sigma\rightarrow\max\\{\frac{1}{2}\delta,\delta(f)\\}^{+}}+\infty$
as $\Gamma$ is of divergence type [29, Cor 7.3].

Let us prove that $\mathcal{P}_{\theta,2}(f;\sigma)$ is finite whenever
$\sigma\delta>\max\\{\delta(f),\frac{1}{2}\delta\\}$. Assume
$\delta(f)>\frac{1}{2}\delta$ (the case $\delta(f)\leq\frac{1}{2}\delta$
follows from a similar approach together with remark 2.12). Let $\sigma$ such
that $\sigma\delta>\delta(f)$, $\varepsilon$ such that
$(\sigma-3\varepsilon)\delta>\delta(f)$ and $T$ such that $\theta(u+t)\leq
e^{\delta\varepsilon u}\theta(t)$ for all $u\geq 0$ and $t\geq T$ as in Lemma
3.5. Using Lemma 2.11 and its notation:

$\displaystyle\mathcal{P}_{\theta,2}$
$\displaystyle(f;\sigma)=\sum_{\Gamma\times\Gamma}f(g^{-1}h)\theta(d_{\Gamma}(g,e))e^{-\sigma\delta
d_{\Gamma}(g,e)}\theta(d_{\Gamma}(h,e))e^{-\sigma\delta d_{\Gamma}(h,e)}$
$\displaystyle=_{h^{\prime}=g^{-1}h}\sum_{h^{\prime}\in\Gamma}f(h^{\prime})\theta(d_{\Gamma}(h^{\prime},e))e^{-\sigma\delta
d_{\Gamma}(h^{\prime},e)}[\sum_{g\in\Gamma}\theta(d_{\Gamma}(g,e))e^{-\sigma\delta
d_{\Gamma}(g,e)}\pi_{\theta,\sigma}(g){\bf 1}_{h^{\prime}}]$
$\displaystyle=\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{-\sigma\delta
d_{\Gamma}(h,e)}(\sum_{m:Rm\leq
T}+\sum_{m:T<Rm})\mathcal{S}_{\theta,m}(\sigma,\sigma;h)$
$\displaystyle\leq\|\sum_{m:Rm\leq
T}\mathcal{S}_{\theta,m}(\sigma,\sigma;\bullet)\|_{\infty}\mathcal{P}_{\theta}(f,\sigma)$
$\displaystyle+\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{-\sigma\delta
d_{\Gamma}(h,e)}\sum_{m:T<Rm}\mathcal{S}_{\theta,m}(\sigma,\sigma;h)$

It enough to prove that the second term, $I$, is finite.

$\displaystyle I$
$\displaystyle=\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{-\sigma\delta
d_{\Gamma}(h,e)}(\sum_{m:T<Rm\leq
d_{\Gamma}(h,e)}+\sum_{m:\max\\{T,d_{\Gamma}(h,e)\\}<Rm})\mathcal{S}_{\theta,m}(\sigma,\sigma;h)$
$\displaystyle\prec\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{-(\sigma-\varepsilon)\delta
d_{\Gamma}(h,e)}\sum_{m:T<Rm\leq d_{\Gamma}(h,e)}e^{2\varepsilon\delta Rm}$
$\displaystyle+\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{(\sigma-1-\varepsilon)\delta
d_{\Gamma}(h,e)}\sum_{m:\max\\{T,d_{\Gamma}(h,e)\\}<Rm}e^{(1+2\varepsilon-2\sigma)\delta
Rm}$ $\displaystyle\prec\frac{1}{e^{2\varepsilon\delta
R}-1}\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{-(\sigma-3\varepsilon)\delta
d_{\Gamma}(h,e)}$ $\displaystyle+\frac{1}{1-e^{(1+2\varepsilon-2\sigma)\delta
R}}\sum_{h\in\Gamma}f(h)\theta(d_{\Gamma}(h,e))e^{-(\sigma-\varepsilon)\delta
d_{\Gamma}(h,e)}$ $\displaystyle=\frac{1}{e^{2\varepsilon\delta
R}-1}\mathcal{P}_{\theta}(f,\sigma-3\varepsilon)+\frac{1}{1-e^{(1+2\varepsilon-2\sigma)\delta
R}}\mathcal{P}_{\theta}(f,\sigma-3\varepsilon)$

that is finite. ∎

######  Remark 3.4.

Note that $\mathcal{P}_{\theta=1,2}(\phi)$ diverges at $\frac{1}{2}\delta$
whenever $\delta(f)\leq\frac{1}{2}\delta$, in other words no rearrangement is
needed.

######  Remark 3.5.

Given a sequence of pointwise positive functions on $\Gamma$, $(f_{n})_{n}$,
with $\lim_{n}\delta(f_{n})=\delta_{\infty}$ finite, $(\sigma_{n})_{n}$ a non-
increasing sequence such that $\sigma_{n}\delta>\delta(f_{n})$ and
$\sigma_{n}\delta\rightarrow\delta_{\infty}$ and $\theta$ has in Lemma . It
follows from Proposition 3.3 that the sequence is well defined

$\mathcal{P}_{\theta,2}(f_{n};\sigma_{n})\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{g,h\in\Gamma}f_{n}(g^{-1}h)\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}\theta(d_{\Gamma}(h,e))e^{-\sigma_{n}\delta d_{\Gamma}(h,e)}$

and

$\mathcal{P}_{\theta,2}(f_{n};\sigma_{n})\xrightarrow{\sigma\rightarrow\max\\{\frac{1}{2}\delta,\delta_{\infty}\\}^{+}}+\infty$

###  3.2. A revisited Gelfand-Naimark-Segal construction

Let
${\boldsymbol{\phi}}=(\phi_{n}\simeq[\pi_{\phi_{n}},\mathcal{H}_{\phi_{n}},{\bf
1}_{\phi_{n}}])_{n}$ in $\mathbb{P}_{+}(\Gamma)$ such that
$\lim_{n}\delta(\phi_{n})=\delta({\boldsymbol{\phi}})\leq\delta$ is well
defined. Let $(\sigma_{n})_{n}$ be a non-increasing sequence such that
$\sigma_{n}\delta>\delta(\phi_{n})$ and
$\sigma_{n}\delta\rightarrow\delta({\boldsymbol{\phi}})$ and $\theta$ as in
Lemma 3.5. As a consequence of Lemma 2.15 the sum

$\sum_{g}\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}\pi_{\phi_{n}}(g){\bf 1}_{\phi_{n}}$

is well defined in $\mathcal{H}_{\phi_{n}}$ if and only if

$\sup_{m}\sum_{g,h\in F_{m}}\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}\theta(d_{\Gamma}(h,e))e^{-\sigma_{n}\delta
d_{\Gamma}(h,e)}\phi_{n}(g^{-1}h)$

is finite for some exhaustion, $(F_{m})_{m}$, of $\Gamma$ by finite sets which
is equivalent in the case $\delta_{\infty}\geq\frac{1}{2}\delta$, using
Proposition 3.3, to

$\mathcal{P}_{\theta}(\phi_{n};\sigma_{n})=\sum_{g\in\Gamma}\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}\phi_{n}(g)$

is finite.

It follows from Subsection 2.5 that the maps:

$J_{n}:({\mathcal{C}}(\overline{\Gamma}),\|\bullet\|_{\infty})\rightarrow\mathcal{H}_{\phi_{n}};\quad
f\mapsto\frac{1}{\sqrt{\mathcal{P}_{\theta,2}(\phi_{n};\sigma_{n})}}\sum_{z}e^{-\sigma_{n}\delta
d_{\Gamma}(z,e)}\theta(d_{\Gamma}(z,e))f(z)\pi_{\phi_{n}}(z){\bf
1}_{\phi_{n}}$

for $n\geq 0$, are well defined bounded intertwiners between the
representations

$\pi_{\theta,\sigma_{n}}(g)f(z)=\frac{\theta(d_{\Gamma}(g,z))}{\theta(d_{\Gamma}(z,e))}e^{-\sigma_{n}\delta
b_{z}(g)}f(g^{-1}z)$

on ${\mathcal{C}}(\overline{\Gamma})$ and $\pi_{\phi_{n}}$.

The pre-unitary structure on ${\mathcal{C}}(\overline{\Gamma})$ is therefore
given by the scalar products

$\displaystyle B_{n}(f)=\|J_{n}(f)\|^{2}$
$\displaystyle=\frac{1}{\mathcal{P}_{\theta,2}(\phi_{n};\sigma_{n})}\sum_{g,h\in\Gamma}f(g)\overline{f(h)}\phi_{n}(g^{-1}h)[\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}][\theta(d_{\Gamma}(h,e))e^{-\sigma_{n}\delta
d_{\Gamma}(h,e)}]$ $\displaystyle=\iint f(g)\overline{f(h)}dm_{n}(g,h)$

where $m_{n}$ is a probability measure defined on the compact space
$\overline{\Gamma}\times\overline{\Gamma}$ as

$m_{n}(F)=\frac{1}{\mathcal{P}_{\theta,2}(\phi_{n};\sigma_{n})}\sum_{\Gamma\times\Gamma}F(g,h)\phi_{n}(g^{-1}h)\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}\theta(d_{\Gamma}(h,e))e^{-\sigma_{n}\delta d_{\Gamma}(h,e)}$

for $F\in{\mathcal{C}}(\overline{\Gamma}\times\overline{\Gamma})$.

###  3.3. A measure theoretic perspective

In order to investigate the quadratic form $B_{n}$ introduced in Subsection
3.2 we consider $m$ a weak limit of sequence $(m_{n})_{n}$ in
$\text{Prob}(\overline{\Gamma}\times\overline{\Gamma})$.

######  Proposition 3.6.

The measure $m$ is invariant by the flip of coordinates (i.e symmetric),
$\Gamma$-quasi-invariant, supported on
${\partial}\Gamma\times{\partial}\Gamma$ and satisfies

$\frac{dg_{*}m}{dm}(\xi,\eta)=e^{-\delta({\boldsymbol{\phi}})b_{\xi}(g^{-1})}e^{-\delta({\boldsymbol{\phi}})b_{\eta}(g^{-1})}$

for all $g\in\Gamma$. Moreover $m(\text{Diag}({\partial}\Gamma))=0$ whenever
$\delta({\boldsymbol{\phi}})>\frac{1}{2}\delta$.

In order to prove Proposition 3.6 we need the following Lemma:

######  Lemma 3.7.

Let $(m_{n})$ and $(m_{n}^{\prime})_{n}$ be two sequences of probability
measure on a compact metric space $Z$ such that $m_{n}\rightarrow m$ and
$m_{n}^{\prime}\rightarrow m^{\prime}$ for the weak topology. Assume $m_{n}$
is absolutely continuous with respect to $m_{n}^{\prime}$ and the Radon-
Nikodym derivatives $f_{n}=\frac{dm_{n}}{dm_{n}^{\prime}}$ are continuous and
converge uniformly to a function $f$. Then $m$ is absolutely continuous with
respect to $m^{\prime}$ and $f=\frac{dm}{dm^{\prime}}$.

###### Proof.

As $f\in{\mathcal{C}}(Z)$ for any $F\in{\mathcal{C}}(Z)$, there exists $N$
such that $\|f-f_{n}\|_{\infty}<\varepsilon\|F\|^{-1}_{\infty}$ and
$|m(fF)-m_{n}(fF)|<\varepsilon$ for all $n\geq N$. It follows

$\displaystyle|m(fF)-m_{n}^{\prime}(F)|$ $\displaystyle=|m(fF)-m_{n}(f_{n}F)|$
$\displaystyle\leq|m_{n}(|f_{n}-f|.|F|)|+|m(fF)-m_{n}(fF)|$
$\displaystyle\leq\|F\|_{\infty}\|f_{n}-f\|_{\infty}+|m(fF)-m_{n}(fF)|\leq
2\varepsilon$

In other words $m_{n}^{\prime}(F)\rightarrow m(fF)$ and thus
$m(fF)=m^{\prime}(F)$ for all $F\in{\mathcal{C}}(Z)$. ∎

###### Proof of Proposition 3.6.

The symmetry follows from the construction. Let
$(\pi_{\phi},\mathcal{H}_{\phi},{\bf 1}_{\phi})$ be the positive cyclic
representation associated to $\phi$.

Given a neighbourhood, $W$, of ${\partial}\Gamma\times{\partial}\Gamma$ one
can find a finite set, $F$, of $\Gamma$ such that
$W^{c}\subset\overline{\Gamma}\times F\cup F\times\overline{\Gamma}$. Indeed
otherwise one can construct a sequence $(x_{n},y_{n})\notin W$ with
$x_{n},y_{n}\rightarrow{\partial}\Gamma$.

Let $F\in{\mathcal{C}}(\overline{\Gamma}\times\overline{\Gamma})$ with
$\|F|_{W}\|_{\infty}\leq\varepsilon$, then

$\displaystyle|m_{n}(F)-m_{n}(F|_{W})|\leq\|F\|_{\infty}m_{n}(\overline{\Gamma}\times
F\cup F\times\overline{\Gamma})$
$\displaystyle\leq\frac{2\|F\|_{\infty}}{\mathcal{P}_{\theta,2}(\phi_{n};\sigma_{n})}\sum_{F}\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}(\pi_{\phi_{n}}(g){\bf
1}_{\phi_{n}},\sum_{\Gamma}\theta(d_{\Gamma}(h,e))e^{-\sigma_{n}\delta
d_{\Gamma}(h,e)}\pi_{\phi_{n}}(h){\bf 1}_{\phi_{n}})$
$\displaystyle\leq_{\text{Cauchy-
Schwarz}}\frac{2\|F\|_{\infty}}{\sqrt{\mathcal{P}_{\theta,2}(\phi_{n};\sigma_{n})}}\sum_{g\in
F}\theta(d_{\Gamma}(g,e))e^{-\sigma_{n}\delta
d_{\Gamma}(g,e)}\xrightarrow{n\rightarrow+\infty}0$

as $F$ is finite and
$\mathcal{P}_{\theta,2}(\phi_{n};\sigma_{n})\rightarrow+\infty$. It follows
that $m$ is supported in ${\partial}\Gamma\times{\partial}\Gamma$.

Using strong hyperbolicity the functions

$D_{\sigma}(\gamma;g,h)=\frac{\theta(d_{\Gamma}(g,\gamma))}{\theta(d_{\Gamma}(g,e))}e^{-\sigma\delta
b_{g}(\gamma,e)}\frac{\theta(d_{\Gamma}(h,\gamma))}{\theta(d_{\Gamma}(h,e))}e^{-\sigma\delta
b_{h}(\gamma,e)}$

on $\Gamma\times\Gamma$ extend continuously to
$\overline{\Gamma}\times\overline{\Gamma}$ as

$D_{\sigma}(\gamma;\xi,h)=\begin{cases}e^{-\sigma\delta
b_{\xi}(\gamma,e)}\frac{\theta(d(h,\gamma))}{\theta(d(h,e))}e^{-\sigma\delta
b_{h}(\gamma,e)}&\quad\text{on ${\partial}\Gamma\times\Gamma$}\\\
e^{-\sigma\delta b_{\xi}(\gamma,e)}e^{-\sigma\delta
b_{\eta}(\gamma,e)}&\quad\text{on
${\partial}\Gamma\times{\partial}\Gamma$.}\end{cases}$

A direct computation shows that

$\frac{d\gamma_{*}m_{n}}{dm_{n}}(g,h)=D_{\sigma_{n}}(\gamma;g,h)$

on $\Gamma\times\Gamma$ for all $\gamma\in\Gamma$.

The sequences of probability measures $(m_{n})_{n}$ and
$(\gamma_{*}m_{n})_{n}$ converge respectively to $m$ and $\gamma_{*}m$.
Moreover $D_{\sigma_{n}}(\gamma;)$ converges uniformly to
$D_{s({\boldsymbol{\phi}})}(\gamma;\bullet)$ for all $\gamma\in\Gamma$. It
follows that $m$ is $\Gamma$-quasi-invariant and

$\frac{dg_{*}m}{dm}(\xi,\eta)=e^{-\sigma\delta
b_{\xi}(\gamma,e)}e^{-\sigma\delta b_{\eta}(\gamma,e)}$

as $m$ is supported on ${\partial}\Gamma\times{\partial}\Gamma$.

Assume $s>\frac{1}{2}$ and $m(\text{Diag}({\partial}\Gamma))\neq 0$, as $m$ is
quasi-invariant and ${\partial}\Gamma$ minimal
$\frac{1}{m(\text{Diag}({\partial}\Gamma))}m|_{{\partial}\Gamma}$ is a
$2s\delta$-conformal probabilities supported on ${\partial}\Gamma$ with
$2s\delta>\delta$ which contradict the uniqueness of the conformal measure for
hyperbolic groups (see [29, Thm. 7.7]) . ∎

Note that $f\mapsto m(f\otimes f)^{\frac{1}{2}}$ is a $\pi_{s}$-invariant
(with $s\delta=s({\boldsymbol{\phi}})\delta=\delta({\boldsymbol{\phi}})$)
semi-norm on ${\mathcal{C}}({\partial}\Gamma)$ and thus satisfies the triangle
inequality. Together with the conformal properties of $m$ it allows us to
recover a partial shadow Lemma:

######  Lemma 3.8 (Upper shadow lemma).

Given $f\in{\mathcal{C}}({\partial}\Gamma)$ a function supported on a Shadow
$\mathcal{O}(\gamma,r_{0})$. Then

$m(f\otimes f)^{\frac{1}{2}}\prec
e^{2\delta({\boldsymbol{\phi}})r_{0}}e^{-\delta({\boldsymbol{\phi}})d_{\Gamma}(\gamma,e)}\|f\|_{\infty}$

This Lemma is used in Theorem’s 8 proof.

###### Proof.

For $\xi\in\mathcal{O}(\gamma,r_{0})$ one has:

$\displaystyle b_{\gamma^{-1}\xi}(\gamma^{-1},e)+d_{\Gamma}(\gamma,e)$
$\displaystyle=-b_{\xi}(\gamma,e)+d_{\Gamma}(\gamma,e)$
$\displaystyle=2(\xi,\gamma)\geq 2d_{\Gamma}(\gamma,e)-2r_{0}$

and thus $b(\gamma^{-1},e)\geq d_{\Gamma}(\gamma,e)-2r_{0}$ on
$\gamma^{-1}\mathcal{O}(\gamma,r_{0})$. It follows

$\displaystyle m(f{\bf 1}_{\mathcal{O}(\gamma,r_{0})}{\otimes}f{\bf
1}_{\mathcal{O}(\gamma,r_{0})})^{\frac{1}{2}}$
$\displaystyle=m((\pi_{s}(\gamma^{-1})f).{\bf
1}_{\gamma^{-1}\mathcal{O}(\gamma,r_{0})}{\otimes}(\pi_{s}(\gamma^{-1})f).{\bf
1}_{\gamma^{-1}\mathcal{O}(\gamma,r_{0})})^{\frac{1}{2}}$ $\displaystyle\leq
e^{2\delta({\boldsymbol{\phi}})r_{0}}e^{-\delta({\boldsymbol{\phi}})d_{\Gamma}(\gamma,e)}m(g^{*}f{\otimes}g^{*}f)^{\frac{1}{2}}$
$\displaystyle\leq
e^{2\delta({\boldsymbol{\phi}})r_{0}}e^{-\delta({\boldsymbol{\phi}})d_{\Gamma}(\gamma,e)}\|f\|_{\infty}$

∎

The $\Gamma$-invariant coordinate projection maps
$p_{i}:{\partial}\Gamma\times{\partial}\Gamma\rightarrow{\partial}\Gamma$
($i=1,2$) induce a disintegration of the $\Gamma$-quasi-invariant probability
measure $m$ over the compact set ${\partial}\Gamma$. As $m$ is symmetric we
denote $(p_{1})_{*}m\cong_{\text{isom.}}(p_{2})_{*}m=\nu$.

As a consequence of the desintegration theorem (see [34, Chapter iii] or [4])
there exist a family on probability measures,
$(\rho_{\xi})_{\xi\in{\partial}\Gamma}$ on ${\partial}\Gamma$ and a
multiplicative cocycle
$c:\Gamma\times{\partial}\Gamma\rightarrow{\mathbb{R}}_{+}$ such that for any
positive Borel function $\varphi$ on ${\partial}\Gamma$
$\xi\mapsto\rho_{\xi}(\varphi)$ is also Borel,

$\int\varphi(\gamma^{-1}\eta)c(\gamma,\eta)d\rho_{\xi}(\eta)=\int\varphi(\eta)d\rho_{\gamma^{-1}\xi}(\eta)$

, i.e the probability $\gamma_{*}\rho_{\xi}$ and $\rho_{\gamma^{-1}\xi}$ are
equivalent, and

$\iint_{{\partial}\Gamma\times{\partial}\Gamma}\Phi
dm=\int_{{\partial}\Gamma}\int_{{\partial}\Gamma}\Phi(\xi,\eta)d\rho_{\xi}(\eta)d\nu(\xi)$

for all positive measurable function $\Phi\in
L^{0}_{+}({\partial}\Gamma\times{\partial}\Gamma)$.

The family $(\rho_{\xi})_{\xi}$ induces $\Gamma$-equivariant conditional
expectation, $E$, on $({\partial}\Gamma,\nu)$ given by

$E(\Phi)_{\xi}=\int_{{\partial}\Gamma}\Phi(\xi,\eta)d\rho_{\xi}(\eta)$

for all positive Borel function on ${\partial}\Gamma\times{\partial}\Gamma$
such that

$\int f(\xi)f^{\prime}(\eta)dm(\xi,\eta)=\int
f(\xi)E_{\xi}(f^{\prime})d\nu(\xi)$

for all $f,f^{\prime}\in L^{0}_{+}({\partial}X)$. In particular $E$ induces a
contraction on every $L^{p}(\nu)$ for $1\leq p\leq\infty$ and satisfies the
monotone convergence theorem.

######  Definition 3.9.

Given $\mu$ a quasi-invariant probability measure on $\overline{\Gamma}$, a
conditional expectation, $E$, is called $(\Gamma,s,\mu)$-conformal if

$E(\pi_{s}(g)f)=\pi_{s}^{*}(g)E(f)$

for $f\in L^{2}(\overline{\Gamma})$ and $\pi_{s}^{*}(g)$ the adjoint of
$\pi_{s}(g)$ in $L^{2}(\overline{\Gamma})$.

######  Proposition 3.10 (Generalized Knapp-Stein relation).

For all $1\leq p\leq\infty$, the conditional expectation $E$ induces a
contraction on $L^{p}({\partial}\Gamma,\nu)$ that is
$(\Gamma,s,\mu)$-conformal. Moreover $E$ is positive definite on
$L^{2}({\partial}\Gamma,\nu)$.

###### Proof.

The contraction property and the positivity on $L^{2}({\partial}\Gamma,\nu)$
follows respectively from general facts about conditional expectations [34]
and the origin of the measure $m$.

A direct computation shows:

$\pi_{s({\boldsymbol{\phi}})}^{*}(g)\psi(\xi)=e^{\delta({\boldsymbol{\phi}})b_{\xi}(g)}\frac{dg^{-1}_{*}\nu}{d\nu}(\xi)\psi(g^{-1}\xi)$

Given $\psi_{1},\psi_{2}\in L^{2}({\partial}\Gamma)$ one has:

$\displaystyle\int\psi_{1}(\xi)\int
e^{-\delta({\boldsymbol{\phi}})b_{\eta}(g)}\psi_{2}(g^{-1}\eta)d\rho_{\xi}(\eta)d\nu(\xi)$
$\displaystyle=\int\psi_{1}(g\xi)e^{\delta({\boldsymbol{\phi}})b_{\eta}(g^{-1})}\psi_{2}(\eta)d(g^{-1})_{*}m(\xi,\eta)=\int
e^{-\delta({\boldsymbol{\phi}})b_{\xi}(g^{-1})}\psi_{1}(g\xi)E(\psi_{2})(\xi)d\nu(\xi)$
$\displaystyle=\int\psi_{1}(\xi)\frac{dg^{-1}_{*}\nu}{d\nu}(\xi)e^{\delta({\boldsymbol{\phi}})b_{\xi}(g)}E(\psi_{2})(g^{-1}\xi)d\nu(\xi)$

In other words

$E(e^{-\delta({\boldsymbol{\phi}})b(g)}g^{*}\psi)(\xi)=e^{\delta({\boldsymbol{\phi}})b_{\xi}(g)}\frac{dg^{-1}_{*}\nu}{d\nu}(\xi)E(\psi)(g^{-1}\xi)$

for all $g\in\Gamma$ and $\psi\in{\mathcal{C}}({\partial}\Gamma)$.

∎

######  Remark 3.11.

Note that

$\frac{dg_{*}\nu}{d\nu}(\xi)=e^{-\delta({\boldsymbol{\phi}})b_{\xi}(g)}E(e^{-\delta({\boldsymbol{\phi}})b_{\bullet}(g)})(\xi)$

$\nu$-almost surely and thus

$E(\pi_{s}(g)\psi)=E(e^{-\delta({\boldsymbol{\phi}})b_{\bullet}(g)})E(\psi)$

for all $\psi\in L^{1}(\nu)$ and $g\in\Gamma$.

###  3.4. Back to quadratic forms

It follows from Subsection 3.3 that
$B_{\boldsymbol{\phi}}^{\partial}(\psi)\stackrel{{\scriptstyle\text{def}}}{{=}}(E(\psi),\psi)_{L^{2}(\nu)}$
is a positive $\pi_{s}$-invariant (with
$s\delta=s({\boldsymbol{\phi}})\delta=\delta({\boldsymbol{\phi}})$) quadratic
form on $L^{2}({\partial}\Gamma,\nu)$.

Observe that the operators
$J_{n}:{\mathcal{C}}(\overline{\Gamma})\rightarrow\mathcal{H}_{\phi_{n}}$
(introduced in Subsection 3.2) induce the maps
$E_{n}=J_{n}^{*}J_{n}:{\mathcal{C}}(\overline{\Gamma})\rightarrow{\mathcal{C}}(\overline{\Gamma})^{\prime}$
such that $(E_{n}(f_{1}),f_{2})=B_{n}(f_{1},f_{2})$ for all
$f_{1},f_{2}\in{\mathcal{C}}(\overline{\Gamma})$.

######  Lemma 3.12.

The sequence of operators $(E_{n})_{n}$ converges to $E$ for weak operator
topology (see Appendix B). In particular

$B_{n}(f_{1},f_{2})\rightarrow
B_{\boldsymbol{\phi}}^{\partial}(f_{1}|_{{\partial}\Gamma},f_{2}|_{{\partial}\Gamma})$

for all $f_{1},f_{2}\in{\mathcal{C}}(\overline{\Gamma})$.

###### Proof.

Given $f_{1},f_{2}\in{\mathcal{C}}(\overline{\Gamma})$ one has:

$\displaystyle\lim_{n}(E_{n}(f_{1}),f_{2})$ $\displaystyle=\iint
f_{1}(x)\overline{f_{2}}(y)dm_{n}(x,y)$ $\displaystyle\rightarrow\iint
f_{1}(\xi)\overline{f_{2}}(\eta)dm(\xi,\eta)=\int_{{\partial}\Gamma}f_{1}(\xi)E(f_{2})_{\xi}d\nu(\xi)$

∎

As $\pi_{\theta,\sigma_{n}}(g)f$ converges uniformly to $\pi_{s}(g)f$ for all
$f\in{\mathcal{C}}(\overline{\Gamma})$ it follows

$(E_{n}(\pi_{\theta,\sigma_{n}}(g)f_{1}),f_{2})\rightarrow(E(\pi_{s}(g)f_{1}),f_{2})$

for all $f_{1},f_{2}\in{\mathcal{C}}(\overline{\Gamma})$ (see Lemma B.1
Appendix B).

##  4\. First properties

Given $\phi\in\mathbb{P}_{+}(\Gamma)$ and $(\pi_{\phi},\mathcal{H}_{\phi},{\bf
1}_{\phi})$ its GNS triple, we denote
$(\pi_{\phi}^{\partial},\mathcal{H}_{\phi}^{\partial},{\bf 1})$ a
$\phi$-boundary representation constructed in Section 3, $B_{n}$ with $n\geq
0$ and $B_{\phi}^{\partial}$ the quadratic forms considered respectively
Subsections 3.2 and 3.4 (in this case $\phi_{n}=\phi$ for all $n\geq 0$ and
$\delta({\boldsymbol{\phi}})=\delta(\phi)$). We assume $\theta=1$ (see Lemma
3.5) whenever no particular adjustment are needed.

###  4.1. The relations between $\pi_{\phi}^{\partial}$ and $\pi_{\phi}$

A representation is mixing if its matrix coefficients belongs to
$c_{0}(\Gamma)$. The mixing property is invariant by isomorphism.

######  Lemma 4.1.

The representation $\pi_{\phi}^{\partial}$ factorises through the kernel of
$\phi$. In particular $\pi_{\phi}^{\partial}$ is not mixing whenever $\phi$
has an infinite kernel.

###### Proof.

For $\sigma>s(\phi)$, $g\in\text{Ker}(\pi_{\phi})$ and
$f_{1},f_{2}\in{\mathcal{C}}(\overline{\Gamma})$ one has:

$\displaystyle
B_{n}(\pi_{\sigma}(\gamma)f_{1},f_{2})=\sum_{g,h}e^{-\sigma\delta
b_{g}(\gamma)}f_{1}(\gamma^{-1}g)f_{2}(h)\phi(g^{-1}h)e^{-\sigma\delta
d_{\Gamma}(g,e)}e^{-\sigma\delta d_{\Gamma}(h,e)}$
$\displaystyle=\sum_{g,h}f_{1}(g)f_{2}(h)\phi(g^{-1}h[h^{-1}\gamma^{-1}h])e^{-\sigma\delta
d(g,e)}e^{-\sigma\delta d(h,e)}$ $\displaystyle=B_{n}(f_{1},f_{2})$

On the other hand, using Lemma 3.12:

$B_{n}(\pi_{\sigma_{n}}(g)(f_{1}),f_{2})=(E_{n}\pi_{\sigma_{n}}(g)(f_{1}),f_{2})\rightarrow(E(\pi_{s}(g)f_{1}),f_{2})$

It follows that $(E(\pi_{s}(g)f_{1}),f_{2})=(E(f_{1}),f_{2})$ and thus
$\pi_{\phi}^{\partial}(g)=1$ for all $g\in\text{Ker}(\pi_{\phi})$ as
${\mathcal{C}}(\overline{\Gamma})$ generates a dense subspace in
$\mathcal{H}_{\phi}^{\partial}$.

∎

######  Remark 4.2.

All the boundary representations considered in [8] [41] [15] but also every
spherical representations of semi-simple Lie groups with finite center are
mixing (due to Howe-Moore theorem [12]). It follows that
$\pi_{\phi}^{\partial}$ does not identify with one of them whenever $\phi$ has
an infinite kernel, in other words $\pi_{\phi}^{\partial}$ is of exotic type.

A unitary representation $\pi$ is weakly contained in $\pi^{\prime}$ if every
$\pi$-matrix coefficient is a limit for the compact-open topology of
$\pi^{\prime}$-matrix coefficients and denoted $\pi\prec\pi^{\prime}$ (see
[10] and [11] appendix F for more about weak containment).

######  Lemma 4.3.

Let ${\boldsymbol{\phi}}=(\phi_{n})_{n}$ be a sequence of positive cyclic
representations such that
$\lim_{n}\delta(\phi_{n})=\delta({\boldsymbol{\phi}})$ is well defined and
$\pi_{\boldsymbol{\phi}}^{\partial}$ an associated boundary representation.
Then $\pi_{n}\rightarrow\pi_{\boldsymbol{\phi}}^{\partial}$ for the Fell
topology. In particular given $\phi\in\mathbb{P}_{+}(\Gamma)$,
$\pi_{\phi}^{\partial}$ is weakly contained in $\pi_{\phi}$ and
$\|\pi_{\phi}^{\partial}(f)\|\leq\|\pi_{\phi}(f)\|$ for all
$f\in{\mathbb{C}}[\Gamma]$.

###### Proof.

Let $[\pi_{n},v_{n}]$ be the positive cyclic representation associated to
$\phi_{n}$. As explained in Subsection 3.2
$\pi_{\theta,\sigma_{n}}\simeq_{\text{isom.}}\pi_{n}$ for all
$n\in{\mathbb{N}}$ and as the $\pi^{\partial}$ matrix coefficients are limits
of $(\pi_{\theta,\sigma_{n}})_{n}$ coefficients due to Lemma 3.12 (and [11,
Lem. F.2.2]) it follows that
$\pi_{\sigma_{n},\theta}\simeq\pi_{n}\rightarrow\pi^{\partial}$. ∎

This can be improved assuming the Poincaré series $\mathcal{P}(\phi)$ and
$\mathcal{P}_{2}(\phi)$ (see Subsection 3.1) satisfy a certain inequality:
As the space $\text{B}_{\leq 1}({\mathcal{C}}(\overline{\Gamma}),\mathcal{H})$
of bounded operators with norms at most $1$ is compact for the weak operator
topology (see Appendix B) one can assume that the sequence
$(J_{\theta,\sigma_{n}})_{n}$ in $\text{B}_{\leq
1}({\mathcal{C}}(\overline{\Gamma}),\mathcal{H})$ introduced in Subsection 3.2
converges to an operator $J_{\phi}$ for this topology. In order to prove that
$J$ is non-zero it is enough to have

$(J_{\theta,\sigma_{n}}{\bf 1}_{\overline{\Gamma}},{\bf
1}_{\phi})=\frac{\mathcal{P}_{\theta}(\phi;\sigma_{n})}{\sqrt{\mathcal{P}_{\theta,2}(\phi;\sigma_{n})}}\geq
c>0$

for $n$ large enough. We call this condition Poincaré sphericity. This leads
to the following criterion:

######  Proposition 4.4.

If $\phi$ is Poincaré spherical then there exists a non-trivial intertwiner
between $\pi_{\phi}^{\partial}$ and $\pi_{\phi}$.

###### Proof.

We are going to prove that the limit $J_{\phi}$ is an intertwiner. Given
$f\in{\mathcal{C}}(\overline{\Gamma})$

$\displaystyle(J_{\theta,\sigma_{n}}\pi_{\theta,\sigma_{n}}(g)f,v)=(J_{\theta,\sigma_{n}}f,\pi_{\phi}(g^{-1})v)$
$\displaystyle\rightarrow(J_{\phi}f,\pi_{\phi}(g^{-1})v)=(\pi_{\phi}(g)J_{\phi}f,v)$

On the other hand, as $\pi_{\theta,\sigma_{n}}(g)f$ converges uniformly to
$\pi_{s}(g)f$, Lemma B.1 implies:

$\displaystyle(J_{\theta,\sigma_{n}}\pi_{\theta,\sigma_{n}}(g)f,v)\rightarrow(J_{\phi}\pi_{s}(g)f,v)$

Since ${\mathcal{C}}(\overline{\Gamma})$ is
$\pi_{\theta,\sigma_{n}}$-invariant dense subspace of
$\mathcal{H}_{\phi}^{\partial}$ and $(J_{\phi}{\bf 1}_{\overline{\Gamma}},{\bf
1}_{\phi})>0$ it follows that $J_{\phi}$ is a non-zero intertwiner.

∎

The Poincaré sphericity condition can be deduced when $\phi$ is roughly
radial:

######  Proposition 4.5.

Let $f$ be a pointwise positive function and assume that $f$ is
$k_{s}$-roughly radial function with $k_{s}(t)=e^{-(1-s)\delta t}$ for
$\frac{1}{2}\leq s\leq 1$. Then
$\mathcal{P}_{2}(f;\sigma)\asymp\mathcal{P}^{2}(f;\sigma)$ for all
$\sigma>s(f)$.

###### Proof.

On a Gromov hyperbolic groups the functional
$\sigma\mapsto\sum_{g\in\Gamma}e^{-\sigma\delta d_{\Gamma}(g,e)}$ for
$\sigma>1$ has a expansion, when $\sigma$ approaches $1^{+}$, comparable to
$\frac{1}{(\sigma-1)}$ due to pure sphericality [29, Thm. 7.2]. As

$\sum_{g}\phi(g)e^{-(s+\varepsilon)\delta
d_{\Gamma}(g,e)}\asymp\sum_{g}k_{s}(d_{\Gamma}(g,e))e^{-(s+\varepsilon)\delta
d_{\Gamma}(g,e)}\asymp\sum_{g}e^{-(1+\varepsilon)\delta
d_{\Gamma}(g,e)}\asymp\varepsilon^{-1}$

for all $\varepsilon>0$ it follows that $\phi$ is of divergent type. Using
Lemma 2.10, Remark 2.12 and their notations:

$\displaystyle\mathcal{P}_{2}(f;\sigma)$
$\displaystyle=\sum_{g,h}e^{-\sigma\delta d_{\Gamma}(g,e)}e^{-\sigma\delta
d_{\Gamma}(h,e)}f(g^{-1}h)$
$\displaystyle=_{h^{\prime}=g^{-1}h}\sum_{h^{\prime}}f(h^{\prime})e^{-\sigma\delta
d_{\Gamma}(h^{\prime},e)}\sum_{g}e^{-\sigma\delta
d_{\Gamma}(g,e)}\pi_{\sigma}(g){\bf 1}_{h^{\prime}}$
$\displaystyle=\sum_{h}f(h)e^{-\sigma\delta d_{\Gamma}(h,e)}(\sum_{m:Rm\leq
d_{\Gamma}(h,e)}+\sum_{m:d_{\Gamma}(h,e)<Rm})\mathcal{S}_{m}(\sigma,\sigma;h)$
$\displaystyle\asymp\sum_{h\in\Gamma}f(h)e^{-\sigma\delta
d_{\Gamma}(h,e)}(d_{\Gamma}(h,e)+1)\asymp\sum_{h\in\Gamma}e^{-(1+(\sigma-s))\delta
d_{\Gamma}(g,e)}(d_{\Gamma}(h,e)+1)$
$\displaystyle\asymp\frac{1}{(\sigma-s)^{2}}$

The last estimate follows from:

$\displaystyle\sum_{h\in\Gamma}$ $\displaystyle e^{-(1+(\sigma-s))\delta
d_{\Gamma}(g,e)}(d_{\Gamma}(h,e)+1)=\sum_{m=0}^{\infty}\sum_{h\in
C_{m}}e^{-(1+(\sigma-s))\delta d_{\Gamma}(g,e)}(d_{\Gamma}(h,e)+1)$
$\displaystyle\asymp_{(1)}\lim_{N}\frac{-1}{1-e^{-(\sigma-s)\delta
R}}\sum_{m\leq N}(e^{-(\sigma-s)\delta R(m+1)}-e^{-(\sigma-s)\delta Rm})m$
$\displaystyle\asymp_{(2)}\lim_{N}\frac{-1}{1-e^{-(\sigma-s)\delta
R}}e^{-(\sigma-s)\delta R(m+1)}(m+1)+\frac{1}{1-e^{-(\sigma-s)\delta
R}}\sum_{1\leq m\leq N}e^{-(\sigma-s)\delta Rm}$
$\displaystyle\asymp[\frac{1}{1-e^{-(\sigma-s)\delta
R}}]^{2}\asymp_{\sigma\sim s^{+}}\frac{1}{(\sigma-s)^{2}}$

where $(1)$ is due to the pure sphericality of $\Gamma$ and $(2)$ a summation
by part. One has shown that
$\mathcal{P}_{2}(f;\sigma)\asymp\mathcal{P}^{2}(f;\sigma)$ for $\sigma\sim
s(f)^{+}$. ∎

####  4.1.1. About the boundary representation of a boundary representation

Given a $\phi$-boundary representation, $\pi_{\phi}^{\partial}$, we denote
$\Xi_{\phi}^{\partial}=(\pi_{\phi}^{\partial}{\bf 1},{\bf 1})$ its Harish-
Chandra function namely the generating function of the positive cyclic
representation $(\pi_{\phi}^{\partial},\mathcal{H}_{\phi}^{\partial},{\bf
1})$.

######  Proposition 4.6.

The function $\Xi_{\phi}^{\partial}\in\mathbb{P}_{+}(\Gamma)$ is Poincaré
spherical of divergent type and exponent
$\max\\{\delta(\phi),\frac{1}{2}\delta\\}$. In particular there exists a non-
zero intertwiner between $\pi_{\phi}^{\partial}$ to
$\pi_{\Xi_{\phi}^{\partial}}^{\partial}$.

###### Proof.

Recall that the conformal exponent of $\pi_{\phi}^{\partial}$ is equal to
$\max\\{s(\phi),\frac{1}{2}\\}$ due to Proposition 3.3.

Using Proposition 3.10 notations and Corollary 2.13 estimates, as $\nu$ is
supported on ${\partial}\Gamma$, one has:

$\displaystyle\mathcal{P}(\Xi_{\phi}^{\partial};\sigma)=\sum_{g\in\Gamma}e^{-\sigma\delta
d_{\Gamma}(g,e)}\Xi_{\phi}^{\partial}(g)$
$\displaystyle=(E(\mathcal{S}_{\infty}(\sigma,s;\bullet)),{\bf
1})_{L^{2}(\nu)}$ $\displaystyle\asymp_{s}\frac{1}{(\sigma-s)}$

and

$\displaystyle\mathcal{P}_{2}(\Xi_{\phi}^{\partial};\sigma)=(E(\mathcal{S}_{\infty}(\sigma,s;\bullet)),\mathcal{S}_{\infty}(\sigma,s;\bullet))_{L^{2}(\nu)}\asymp_{s}\frac{1}{(\sigma-s)^{2}}$

for $s>\frac{1}{2}$. Similarly for $s=\frac{1}{2}$ :

$\mathcal{P}(\Xi_{\phi}^{\partial};\sigma)\asymp\frac{1}{(\sigma-\frac{1}{2})^{2}}\quad\text{and}\quad\mathcal{P}_{2}(\Xi_{\phi}^{\partial};\sigma)\asymp\frac{1}{(\sigma-\frac{1}{2})^{4}}$

It follows that
$\max\\{\delta(\phi),\frac{1}{2}\delta\\}=\delta(\Xi_{\phi}^{\partial})$,
$\Xi_{\phi}^{\partial}$ is of divergent type and Poincaré spherical. The rest
follows from Proposition 4.4.

∎

######  Remark 4.7.

It is reasonable to conjecture that $\pi_{\phi}^{\partial}$ is irreducible. In
this case Poincaré sphericity implies that $\pi_{\phi}^{\partial}$ is realised
as a subrepresentation of $\pi_{\phi}$. In particular this would imply that
$\pi_{\phi}^{\partial}$ and $\pi_{\Xi_{\phi}^{\partial}}^{\partial}$ are
isomorphic.

###  4.2. Natural complementary series

In certain situations the invariant scalar product, $B_{\phi}^{\partial}$, can
be explicitly described.
Given $t>\delta$ we denote $\mu_{t}$ the probability measure:

$\mu_{t}=\frac{1}{\mathcal{P}({\bf
1}_{\Gamma},\delta^{-1}t)}\sum_{g\in\Gamma}e^{-td_{\Gamma}(g,e)}\text{Dir}_{g}$

on $\overline{\Gamma}$ where $\text{Dir}_{g}$ stands for the Dirac measure at
$g\in\Gamma$. As a consequence of strong hyperbolicity one has
$\mu_{t}\xrightarrow{t\rightarrow\delta^{+}}\nu_{\text{PS}}$ where
$\nu_{\text{PS}}$ stands for the Patterson-Sullivan measure on
${\partial}\Gamma$. Indeed strong hyperbolicity implies that any limit of
$(\mu_{t})_{t>\delta}$ must be a $(\Gamma,d_{\Gamma})$-conformal probability
but this probability is unique (see [29, Thm. 7.7]).

######  Proposition 4.8.

Let $f$ be a pointwise positive function that is $k_{s}$-roughly radial
function with $k_{s}(t)=e^{-(1-s)\delta t}$ ($\frac{1}{2}<s\leq 1$), $m$ and
$\nu$ be the measures considered in Subsection 3.3. Then $f$ is of divergent
type, the measure $\nu$ and $\nu_{\text{PS}}$ are equivalent with
$m=c(s).m_{s}$ where

$m_{s}(\Phi)=\iint\Phi(\xi,\eta)e^{2(1-s)\delta(\xi,\eta)}d\nu_{\text{PS}}(\xi)d\nu_{\text{PS}}(\eta)$

and $c(s)=(m_{s}({\bf 1}_{{\partial}\Gamma\times{\partial}\Gamma}))^{-1}$ for
all $\Phi\in{\mathcal{C}}({\partial}\Gamma\times{\partial}\Gamma)$.

In order to prove Proposition 4.8 we need the following results proved in [15]
(see Lemma 2.4 and Corollary 4.1):

######  Lemma 4.9 (sequential shadow lemma).

There exist $\delta<\delta^{\prime}$ and $r_{0}$ such that:

$\mu_{t}(\\{x\in\overline{\Gamma}:(x,h)\geq d_{\Gamma}(h,e)-r\\})\asymp
e^{tr}e^{-td_{\Gamma}(h,e)}$

for all $\delta<t<\delta^{\prime}$, $h\in\Gamma$ and $r>r_{0}$.

######  Corollary 4.10.

The sequence of probabilities

$dm_{s;t}(g,h)=e^{2(1-s)\delta(g,h)}d\mu_{t}(g)d\mu_{t}(h)$

converges weakly to

$dm_{s}(\xi,\eta)=e^{2(1-s)\delta(\xi,\eta)}d\nu_{\text{PS}}(\xi)d\nu_{\text{PS}}(\eta)$

on $\overline{\Gamma}\times\overline{\Gamma}$.

######  Remark 4.11.

Even if those results was originally stated in a CAT(-1) context their proofs
adapt to proper roughly geodesic strongly hyperbolic spaces.

###### Proof of Proposition 4.8.

It follows from Proposition 4.5 that $f$ is of divergent type and
$\mathcal{P}_{2}(f;\sigma)\asymp\mathcal{P}^{2}(f;\sigma)$ for all $\sigma\sim
s(f)^{+}$.

Given $\Phi\in{\mathcal{C}}_{+}(\overline{\Gamma}\times\overline{\Gamma})$ and
$\sigma>s(f)$ one has:

$\displaystyle\frac{1}{\mathcal{P}_{2}(f;\sigma)}\iint_{\Gamma\times\Gamma}\Phi(g,h)\phi(g^{-1}h)e^{-\sigma\delta
d(g,e)}e^{-\sigma\delta d(h,e)}dhdg$
$\displaystyle\prec\frac{1}{\mathcal{P}_{2}(f;\sigma)}\iint_{\Gamma\times\Gamma}\Phi(g,h)e^{2(1-s)\delta(g,h)_{e}}e^{-(1+\sigma-s)\delta
d(g,e)}e^{-(1+\sigma-s)\delta d(h,e)}dhdg$
$\displaystyle\asymp\iint_{\Gamma\times\Gamma}\Phi(g,h)dm_{s;t}(g,h)$

Using Lemma 4.9 it follows $m(\Phi)\leq m_{s}(\Phi)$ and thus the non-zero
functional

$\varphi:L^{1}(m_{s})\rightarrow{\mathbb{C}};\quad\Phi\mapsto m(\Phi)$

is a well defined, bounded and order preserving operator.

Therefore one can find a measurable positive and essentially bounded function
$F$ on ${\partial}\Gamma\times{\partial}\Gamma$ such that

$m(\Phi)=\varphi(\Phi)=\iint_{{\partial}\Gamma\times{\partial}\Gamma}\Phi(\xi,\eta)F(\xi,\eta)e^{2(1-s)\delta(\xi,\eta)_{e}}d\nu_{\text{PS}}(\xi)d\nu_{\text{PS}}(\eta)$

As $\varphi(\pi_{s}\otimes\pi_{s}(g)\Phi)=\varphi(\Phi)$ for all
$\Phi\in{\mathcal{C}}({\partial}\Gamma\times{\partial}\Gamma)$ it follows that
$g^{*}F=F$ and thus $F$ must be constant as the $\Gamma$-action on
${\partial}\Gamma$ is $2$-ergodic due to [30, Thm. 4.1] (see also [7] or [41,
Appendix A]). ∎

In this case the $(\Gamma,s,\nu_{\text{PS}})$-conditional expectation is the
renormalised Knapp-Stein operator

$\mathcal{I}_{s}^{\prime}(\psi)_{\eta}=(\omega_{1}(\eta))^{-1}\int
f(\xi)e^{2(1-s)(\xi,\eta)}d\nu_{\text{PS}}(\xi)$

on $L^{2}(\omega_{1}d\nu_{\text{PS}})$ with

$\omega(\eta)=\int e^{2(1-s)(\xi,\eta)}d\nu_{\text{PS}}(\xi)$

and $\omega_{1}=\frac{\omega}{\nu_{\text{PS}}(\omega)}$.

It follows from [15, Lem 3.6] that the Harish-Chandra functions:

$\Xi_{s}(g)\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathcal{I}_{s}^{\prime}(\pi_{s}(g){\bf
1}),{\bf 1})_{L^{2}(\omega_{1}d\nu_{\text{PS}})}$

on $\Gamma$ for $\frac{1}{2}<s\leq 1$ satisfy $\Xi_{s}(g)\asymp
e^{-(1-s)\delta d_{\Gamma}(g,e)}$. We deduce, using Proposition 4.8:

######  Corollary 4.12.

The Knapp-Stein of order $\frac{1}{2}<s\leq 1$ is positive definite iff there
exists a positive definite function over $\Gamma$ such that $\phi\asymp
e^{-(1-s)\delta d_{\Gamma}(\bullet,e)}$.

######  Corollary 4.13.

The Knapp-Stein operators are positive definite for all $\frac{1}{2}<s\leq 1$
iff $\Gamma$ has a proper affine action with unitary cocycle $b$ such that
$\|b(g)\|^{2}\simeq_{X}c.d_{\Gamma}(g,e)$ for a fixed $c>0$.

###### Proof.

Indeed if $\Gamma$ admits such a affine action, then
$\phi_{t}(g)\stackrel{{\scriptstyle\text{def}}}{{=}}e^{-t\|b(g)\|^{2}}\asymp
e^{-td_{\Gamma}(g,e)}$ is roughly $k_{t}$-spherical with $k_{t}(x)=e^{-tx}$
with $\phi_{t}\in\mathbb{P}_{+}(\Gamma)$ due to Schoenberg’s theorem [11,
Prop. 2.11.1] and Proposition 4.8 can be applied to prove the positivity of
the Knapp-Stein operator. The converse follows from [16, Thm. 2]. ∎

The following semi-group relation holds between the natural complementary
series:

######  Corollary 4.14.

Given $\frac{1}{2}<s,s^{\prime},t\leq 1$ with $s+s^{\prime}=1+t$ and
$\Xi_{s},\Xi_{s^{\prime}}\in\mathbb{P}_{+}(\Gamma)$, one has
$\Xi_{t}\in\mathbb{P}_{+}(\Gamma)$ and
$\pi_{t}\subset\pi_{s}\otimes\pi_{s^{\prime}}$.

###### Proof.

The first part follows from the estimate $\Xi_{s}.\Xi_{s^{\prime}}(g)\asymp
e^{-(1-t)\delta d_{\Gamma}(g,e)}$ (see [15, Lem 3.6]) for all $g\in\Gamma$ and
Proposition 4.8.

Proposition 4.4 together with Proposition 4.5 imply the existence of a non-
zero intertwiner between $\pi_{t}$ and
$\pi_{\Xi_{s}.\Xi_{s^{\prime}}}\subset\pi_{s}\otimes\pi_{s^{\prime}}$. On the
other hand the irreducibility of $\pi_{t}$ [15, Thm. 1] (see also [22])
implies the injectivity. ∎

###  4.3. About the critical case $\delta=\delta(\phi)$

######  Theorem 8.

Let ${\boldsymbol{\phi}}=(\phi_{n})_{n}\in\mathbb{P}_{+}(\Gamma)$,
$([\pi_{\phi_{n}},{\bf 1}_{\phi_{n}}])_{n}$ the sequence of positive cyclic
representations associated and assume
$\lim_{n}\delta(\phi_{n})=\delta({\boldsymbol{\phi}})=\delta$, then
$\pi^{\partial}\simeq 1$ and $\pi_{n}\rightarrow 1$. In particular given
$\phi\in\mathbb{P}_{+}(\Gamma)$ with $\delta(\phi)=\delta$,
$1\prec\pi_{\phi}$.

The proof of Theorem 8 uses ideas coming from Proposition 5.21 and Corollary
5.22 of [30].

###### Proof.

Let $(\pi^{\partial},\mathcal{H}^{\partial},\|.\|_{s})$ be a boundary
representation associated to $(\phi_{n})_{n}$ and $([\pi_{n},v_{n}])_{n}$ the
cyclic representations associated. As a consequence of Vitali’s Lemma 2.4 for
all $n\in{\mathbb{N}}$, there exists a continuous partition of the unit of
${\partial}\Gamma$, $(\chi_{\gamma})_{\gamma\in C_{n}^{*}}$, with
$C_{n}^{*}\subset C_{n}$ such that

$\mathcal{O}(\gamma,R)\subset\\{\chi_{\gamma}\neq
0\\}\subset\mathcal{O}(\gamma,4R)$

and $\chi_{\gamma}|_{\mathcal{O}(\gamma,R)}=1$ (see Subsection 2.2 for
notations). Given $f\in{\mathcal{C}}({\partial}\Gamma)$ Lemma 3.8 implies:

$\displaystyle\|f\|_{s}\leq\sum_{\gamma\in
C_{n}^{*}}\|f\chi_{\gamma}\|_{s}=\sum_{\gamma\in
C_{n}^{*}}m((f\chi_{\gamma}){\otimes}(f\chi_{\gamma}))^{\frac{1}{2}}\prec\sum_{\gamma\in
C_{n}^{*}}e^{-\delta Rn}f_{\gamma}$

where $f_{\gamma}=\sup_{\mathcal{O}(\gamma,4R)}|f|$.

Using uniform continuity on the compact space ${\partial}\Gamma$, for all
$\varepsilon>0$, there exists $n_{0}$, for all $n\geq n_{0}$ and $\gamma\in
C_{n}^{*}$, $|f_{\gamma}-f|\leq\varepsilon$ on $\mathcal{O}(\gamma,4R)$.

On the other hand

$\displaystyle\int_{{\partial}\Gamma}|f|d\nu_{\text{PS}}\geq\sum_{\gamma\in
C_{n}^{*}}\int_{\mathcal{O}(\gamma,R)}|f|d\nu_{\text{PS}}\succ\sum_{\gamma\in
C_{n}^{*}}e^{-\delta Rn}(f_{\gamma}-\varepsilon)\succ\sum_{\gamma\in
C_{n}^{*}}e^{-\delta Rn}f_{\gamma}-\varepsilon^{\prime}$

and thus $\|f\|_{s}\leq\|f|L^{1}(\nu_{\text{PS}})\|$.

This induces a bounded map:

$T:L^{1}({\partial}\Gamma,\nu_{\text{PS}})\rightarrow\mathcal{H}^{\partial}$

and as Hilbert spaces are reflexive and thus have Radon-Nikodym property (see
[36, Chap. iii] ) it implies the existence of $K_{T}\in
L^{\infty}({\partial}\Gamma,\nu_{\text{PS}};\mathcal{H}^{\partial})$ such that

$T(f)=\int_{{\partial}\Gamma}f(\xi)K_{T}(\xi)d\nu_{\text{PS}}(\xi)\in\mathcal{H}^{\partial}$

in the sense of Bochner integral.

In particular given any bounded map $V:\mathcal{H}^{\partial}\rightarrow B$
with range into a Banach space $B$, one has

$V\circ
T(f)=\int_{{\partial}\Gamma}f(\xi)V(K_{T}(\xi))d\nu_{\text{PS}}(\xi)\in B$

in the sense of Bochner integral (see [36] Chapter ii Theorem 6). It follows:

$\displaystyle(T(f_{1}),T(f_{2}))_{s}=\int_{{\partial}\Gamma}f_{1}(\xi)(K_{T}(\xi),T(f_{2}))_{s}d\nu_{\text{PS}}(\xi)$
$\displaystyle=\int_{{\partial}\Gamma}f_{1}(\xi)\int_{{\partial}\Gamma}f_{2}(\eta)(K_{T}(\xi),K_{T}(\eta))_{s}d\nu_{\text{PS}}(\eta)d\nu_{\text{PS}}(\xi)$
$\displaystyle=\iint_{{\partial}\Gamma\times{\partial}\Gamma}f_{1}(\xi)f_{2}(\eta)\Phi_{T}(\xi,\eta)d\nu_{\text{PS}}(\eta)d\nu_{\text{PS}}(\xi)$

with $\Phi_{T}\in L^{\infty}({\partial}\Gamma\times{\partial}\Gamma)$ and all
$f_{1},f_{2}\in L^{1}({\partial}\Gamma,\nu_{\text{PS}})$.

Note that the monotone class Lemma (see [3, 4.13]) and the finiteness of
$m_{T}=\Phi_{T}\nu_{\text{PS}}\otimes\nu_{\text{PS}}$ implies that $m_{T}$ is
determined by its values on $L^{1}(\nu_{\text{PS}})\times
L^{1}(\nu_{\text{PS}})$.

It follows from the $\pi_{1}$-invariant of the quadratic form $\|.\|_{s}^{2}$
that $\Phi_{T}$ is $\Gamma$-invariant and thus constant, using the
$2$-ergodicity of the measure $\nu_{\text{PS}}\otimes\nu_{\text{PS}}$ [30,
Thm. 4.1]. Eventually

$\|f\|_{s}^{2}=|\int f(\xi)d\nu_{\text{PS}}(\xi)|^{2}$

and using Lemma 4.3 it follows $\pi_{n}\rightarrow 1\simeq\pi^{\partial}$. ∎

Theorem 8 should be compared with [30] Theorem 1.1:
Given a subset $S\subset\Gamma$ the critical exponent of $S$ is defined as

$\limsup_{r}\frac{1}{r}\log|S\cap B_{\Gamma}(e,r)|$

In particular $\delta(S)=\delta({\bf 1}_{S})$.

######  Corollary 4.15.

Let $\Lambda\subset\Gamma$ be a subgroup. If $\delta(\Lambda)=\delta({\bf
1}_{\Lambda})=\delta$ then $\Lambda$ is co-amenable in $\Gamma$.

###  4.4. About the regular representation

The regular representation, $(\lambda,\ell^{2}(\Gamma))$, can be viewed as a
cyclic representation with respect to the vector
$\text{Dir}_{e}\in\ell^{2}(\Gamma)$. In this case $\phi_{\lambda}={\bf 1}_{e}$
and thus $\delta(\phi_{\lambda})=0$. Observe that

$\mathcal{P}_{2}(\phi_{\lambda},\sigma)=\mathcal{P}(\phi_{\lambda},2\sigma)\asymp\frac{1}{2\sigma-1}$

and for all $f\in{\mathcal{C}}(\overline{\Gamma})$:

$\displaystyle
B_{\lambda,\sigma}(f)=\iint|f(g)|^{2}d\mu_{2\sigma}(g)\xrightarrow{\sigma\rightarrow\frac{\delta}{2}^{+}}\|f|L^{2}(\nu_{\text{PS}})\|^{2}$

It follows that the $\phi_{\lambda}$-boundary representation is uniquely
defined and corresponds to the Koopman representation on
$L^{2}({\partial}\Gamma,\nu_{\text{PS}})$.

As a consequence of Lemma 4.3:

######  Corollary 4.16.

Let $(\pi_{\frac{1}{2}},L^{2}({\partial}\Gamma,\nu_{\text{PS}}))$ be the
unitary Koopman representation over the boundary
$({\partial}\Gamma,\nu_{\text{PS}})$, then $\pi_{\frac{1}{2}}\prec\lambda$. In
particular $\|\pi_{\frac{1}{2}}(f)\|\leq\|\lambda(f)\|$ for all
$f\in{\mathbb{C}}[\Gamma]$.

This fact is known by the mean of amenable actions (see [4] for definitions):

######  Theorem (Kuhn [58]).

Let $\Lambda$ be a discrete group, $\Lambda\curvearrowright(Z,\mu)$ an
amenable action and $(\pi_{\mu},L^{2}(Z,\mu))$ the associated Koopman
representation. Then $\pi_{\mu}\prec\lambda$. In particular
$\|\pi_{\mu}(f)\|\leq\|\lambda(f)\|$ for all $f\in{\mathbb{C}}[\Gamma]$.

On the other hand it follows from Adams [1] that the action of a Gromov
hyperbolic group $\Gamma$ on its boundary $({\partial}\Gamma,\mu)$ endowed
with any quasi-invariant measure, $\mu$, is amenable. Corollary 4.16 provides
another perspective on this fact.

##  5\. Spectral bounds

Let $\Lambda$ be a finitely generated group endowed with a $\Lambda$-invariant
distance, $d_{\Lambda}$, quasi-isometric to a word distance.

We start with some elementary properties about critical exponents:

######  Lemma 5.1.

Let $f$ be a pointwise positive function on $\Lambda$.

  1. (1)

$\delta(f)\leq(1-\frac{1}{p})\delta$ whenever $f\in\ell^{p}(\Lambda)$ for
$p\in[1,+\infty]$.

  2. (2)

For all $1\leq p<\infty$, $\delta(f^{p})\leq\delta(f)$ whenever $f$ is
bounded.

  3. (3)

Given two positive finitely supported functions
$h_{1},h_{2}\in{\mathbb{R}}_{+}[\Lambda]$ one has
$\delta(h_{1}*f*h_{2})=\delta(f)$.

###### Proof.

Let $q$ such that $\frac{1}{p}+\frac{1}{q}=1$ and $\sigma$ such that
$q\sigma>1$, i.e $\sigma>1-\frac{1}{p}$. Then

$\sum_{\Gamma}f(g)e^{-\sigma\delta
d_{\Lambda}(g,e)}\leq[\sum_{\Lambda}|f(g)|^{p}]^{\frac{1}{p}}[\sum_{\Lambda}e^{-\sigma
q\delta d_{\Lambda}(g,e)}]^{\frac{1}{q}}$

is finite.

For second part observe that replacing $f$ with $\frac{f}{\|f\|_{\infty}}$ we
can assume $f\leq 1$ and the result is a consequence of $x^{p}\leq x$ on the
interval $[0,1]$ for $1\leq p<\infty$.

The third part follows from:

$\displaystyle\sum_{g\in\Lambda}$ $\displaystyle
h_{1}*f*h_{2}(g)e^{-\sigma\delta d_{\Lambda}(g,e)}=\sum_{\Gamma^{\times
3}}h_{1}(a)f(a^{-1}gb^{-1})h_{2}(b)e^{-\sigma\delta d_{\Lambda}(g,e)}$
$\displaystyle=\sum_{\Lambda^{\times 3}}h_{1}(a)f(g)h_{2}(b)e^{-\sigma\delta
d_{\Lambda}(agb,e)}$
$\displaystyle\leq(\sum_{a\in\Lambda}h_{1}(a)e^{\sigma\delta
d_{\Lambda}(a,e)})(\sum_{b\in\Lambda}e^{\sigma\delta
d_{\Lambda}(b,e)}h_{2}(b))(\sum_{g\in\Lambda}f(g)e^{-\sigma\delta
d_{\Lambda}(g,e)})$ $\displaystyle\prec_{h_{1},h_{2}}\mathcal{P}(f;\sigma)$

∎

###  5.1. Rapid decay and critical exponent

The reader can refer to [28] for more details about the rapid decay property
and extra examples. We denote
$\lambda:\Lambda\rightarrow\text{B}(\ell^{2}(\Lambda))$ the regular
representation of $\Lambda$ on $\ell^{2}(\Lambda)$.

######  Definition 5.2.

A discrete metric group $(\Lambda,d_{\Lambda})$ has the rapid decay (RD)
property, if there exist constants $C\in{\mathbb{R}}_{+}$ and
$m\in{\mathbb{N}}$ such that for all $f\in{\mathbb{C}}[\Lambda]$ supported in
a ball of radius $R$: $\|\lambda(f)\|_{\text{op}}\leq CR^{m}\|f\|_{2}$.

######  Theorem 9 (Jolissaint [50]).

Gromov hyperbolic groups have the rapid decay property.

We are going to use a weaker form of RD, namely the radial rapid decay
property [79]:
A function $f\in{\mathbb{C}}[\Lambda]$ is called radial on
$(\Lambda,d_{\Lambda})$ if $f(g)=k(d_{\Lambda}(g,e))$ for some function $k$.

######  Definition 5.3.

A discrete metric group $(\Lambda,d_{\Lambda})$ has the radial rapid decay
(RRD) property, if there exist constants $C\in{\mathbb{R}}_{+}$ and
$m\in{\mathbb{N}}$ such that for all radial function
$f\in{\mathbb{C}}[\Lambda]$ supported on a ball of radius $R$:
$\|\lambda(f)\|_{\text{op}}\leq CR^{m}\|f\|_{2}$.

######  Remark 5.4.

It was recently proved in [23] that RRD property is strictly weaker than RD.

The RRD property can be used to estimate the critical exponent of certain
representations:

######  Proposition 5.5.

Let $(\Lambda,d_{\Lambda})$ with the radial rapid decay property and
$\phi\in\mathbb{P}_{+}(\Gamma)$. If $\phi\prec\lambda$ then
$\delta(\phi)\leq\frac{\delta}{2}$.

###### Proof.

As $(\Lambda,d_{\Lambda})$ has RRD one can find $C>0$ and $k\in{\mathbb{N}}$
as in Definition 5.3. Note that for all $\kappa>0$, there exists $r_{0}$, for
all $r\geq r_{0}$: $|B(e,r)|\leq e^{(\delta+\kappa)r}$ and thus

$\displaystyle\sum_{g\in B(e,r)}\phi(g)e^{-\sigma\delta d_{\Lambda}(g,e)}$
$\displaystyle=\sum_{g\in B(e,r)}(\pi_{\phi}(g){\bf 1}_{\phi},{\bf
1}_{\phi})e^{-\sigma\delta d_{\Lambda}(g,e)}$
$\displaystyle\leq\|\pi_{\phi}(\sum_{g\in B(e,r)}e^{-\sigma\delta
d_{\Lambda}(g,e)}\text{Dir}_{g})\|_{\text{op}}$
$\displaystyle\leq_{\phi\prec\lambda}\|\lambda(\sum_{g\in
B(e,r)}e^{-\sigma\delta d_{\Lambda}(g,e)}\text{Dir}_{g})\|_{\text{op}}$
$\displaystyle\leq_{\text{RRD}}Cr^{m}\|\sum_{g\in B(e,r)}e^{-\sigma\delta
d_{\Lambda}(g,e)}\text{Dir}_{g}\|_{2}\prec r^{m}e^{\frac{1}{2}\kappa
r}e^{(\frac{1}{2}-\sigma)\delta r}+1$

for all $r>r_{0}$. It follows for $\sigma>\frac{1}{2}\delta$, $\kappa>0$ small
enough and $r_{0}(\kappa)$ that

$\sum_{g\in\Gamma}\phi(g)e^{-\sigma\delta d_{\Lambda}(g,e)}=\lim_{r}\sum_{g\in
B(e,r)}\phi(g)e^{-\sigma\delta
d_{\Lambda}(g,e)}\prec\lim_{r}r^{k}e^{\frac{1}{2}\kappa
r}e^{(\frac{1}{2}-\sigma)\delta r}+1$

is finite. ∎

######  Remark 5.6.

As a consequence, in the hyperbolic case, the $\phi$-boundary representation
$\pi_{\phi}^{\partial}$ is non-tempered (i.e
$\pi_{\phi}^{\partial}\nprec\lambda$) whenever
$\delta(\phi)>\frac{1}{2}\delta$.
The author don’t know whether $\delta(\phi)\leq\frac{\delta}{2}$ implies
$\pi_{\phi}\prec\lambda$ (or even $\pi_{\phi}^{\partial}\prec\lambda$).

###  5.2. Spectral gap and exponential mixing

We establish an upper bound on the spectral radius of
$\|\pi_{\phi}(f)\|_{\text{op}}$ for $\phi\in\mathbb{P}_{+}(\Lambda)$ and
$f\in{\mathbb{C}}[\Lambda]$ in terms of its Poincaré exponent and regular
spectral radius, namely $\|\lambda(f)\|_{\text{op}}$.

Given $f\in{\mathbb{C}}[\Lambda]$ we denote $r(f)$ the minimal $r>0$ such that
$\\{f\neq 0\\}\subset B(e,r)$.

Theorem 1. Given a positive cyclic representation, $\phi$, on $\Lambda$ one
has:

$\|\pi_{\phi}(f)\|_{\text{op}}\leq
e^{\frac{1}{2}\delta(\phi)r(f)}\|\lambda(f)\|_{\text{op}}$

for all $f\in{\mathbb{C}}[\Lambda]$.

In order to prove Theorem 1 we recall the following technical Lemma’s proved
in [31] (respectively p.101 and p.102):

######  Lemma 5.7.

Given a unitary representation $(\pi,\mathcal{H})$,
$\mathcal{H}_{\infty}\subset\mathcal{H}$ a dense subspace and
$f\in{\mathbb{C}}[\Lambda]$ one has:

$\|\pi(f)\|_{\text{op}}=\sup_{v\in\mathcal{H}_{\infty}}\lim_{n}(\pi(f^{*}*f^{*2n})v,v)^{\frac{1}{4n}}$

######  Lemma 5.8.

For all $f\in{\mathbb{C}}[\Lambda]$,
$\lim_{n}\|(f^{*}*f)^{*2n}\|_{2}^{\frac{1}{4n}}=\|\lambda(f)\|_{\text{op}}$.

###### Proof of Theorem 1.

For all $f\in{\mathbb{C}}[\Lambda]$ and $v=\pi_{\phi}(f){\bf 1}_{\phi}$ one
has $(\pi_{\phi}(g)v,v)=\overline{f}*\phi*f^{\vee}(g)$ with
$f^{\vee}(g)=f(g^{-1})$ and, using Lemma 5.1:

$\delta(|(\pi_{\phi}v,v)|)\leq\delta(|f|*\phi*|f^{\vee}|)\leq\delta(\phi)$

As $(\pi_{\phi},{\bf 1}_{\phi})$ is a cyclic representation the subspace
$\mathcal{H}_{\infty}\stackrel{{\scriptstyle\text{def}}}{{=}}\langle\pi_{\phi}(f){\bf
1}_{\phi}:f\in{\mathbb{C}}[\Lambda]\rangle\subset\mathcal{H}$ is dense.

It follows from Lemma 5.7 that

$\displaystyle\|\pi_{\phi}(f)\|_{\text{op}}=\sup_{v\in\mathcal{H}_{\infty}}\lim_{n}(\sum_{\Lambda}(f^{*}*f)^{*2n}(g)(\pi_{\phi}(g)v,v))^{\frac{1}{4n}}$

for all $f\in{\mathbb{C}}[\Lambda]$.

Let $S=\\{f\neq 0\\}\subset B(e,r(f))$, then
$S^{(n)}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(f^{*}*f)^{*2n}\neq
0\\}\subset B(e,4nr(f))$ and thus, for $s$ such that $s\delta=\delta(\phi)$:

$\displaystyle(\sum_{\Lambda}(f^{*}*f)^{*2n}(g)(\pi_{\phi}(g)v,v))^{2}=(\sum_{S^{(n)}}e^{\frac{s}{2}\delta
d_{\Lambda}(g,e)}(f^{*}*f)^{*2n}(g)e^{-\frac{s}{2}\delta
d_{\Lambda}(g,e)}(\pi_{\phi}(g)v,v))^{2}$ $\displaystyle\leq e^{s\delta
4nr(f)}\|(f^{*}*f)^{*2n}\|_{2}^{2}\sum_{S^{(n)}}e^{-s\delta
d_{\Lambda}(g,e)}|(\pi_{\phi}(g)v,v)|^{2}$ $\displaystyle\leq e^{s\delta
4nr(f)}\|(f^{*}*f)^{*2n}\|_{2}^{2}(\sum_{S^{(n)}}e^{-(1+\frac{\varepsilon}{2})s\delta
d_{\Lambda}(g,e)}|(\pi_{\phi}(g)v,v)|^{2+\varepsilon})^{\frac{2}{2+\varepsilon}}|S^{(n)}|^{\frac{\varepsilon}{2+\varepsilon}}$
$\displaystyle\prec e^{s\delta
4nr(f)}e^{\frac{\varepsilon}{2+\varepsilon}\delta
8nr(f)}\mathcal{P}(|(\pi_{\phi}v,v)|^{2+\varepsilon};s+\frac{\varepsilon
s}{2})\|(f^{*}*f)^{*2n}\|_{2}^{2}$

for any $\varepsilon>0$, where the last inequality uses the fact that:
$|S^{(n)}|\leq|B(e,4nr(f))|\leq Ce^{8\delta nr(f)}$ for some constant $C$.

As a consequence of Lemma 5.1
$\delta(|(\pi_{\phi}v,v)|^{2+\varepsilon})\leq\delta(|(\pi_{\phi}v,v)|)$ and
using Lemma 5.8 :

$\displaystyle\limsup_{n}(\sum_{\Lambda}(f^{*}*f)^{*2n}(g)(\pi(g)v,v))^{\frac{1}{4n}}$
$\displaystyle\leq e^{\frac{\varepsilon}{2+\varepsilon}\delta
r(f)}e^{\frac{1}{2}s\delta r(f)}\|\lambda(f)\|_{\text{op}}$

As $\varepsilon$ can be taken arbitrarily small the inequality follows. ∎

######  Remark 5.9.

The above inequality can be extended to all representation $\pi$ by
considering $\pi\otimes\overline{\pi}$ where $\overline{\pi}$ stands for the
contragredient dual representation of $\pi$ as every vector
$v\otimes\overline{v}$ is positive, i.e
$\phi=(\pi\otimes\overline{\pi}v\otimes\overline{v},v\otimes\overline{v})\in\mathbb{P}_{+}(\Lambda)$.

Observe that given $\phi\in\mathbb{P}_{+}(\Lambda)$ in
$\phi\in\ell^{p+\varepsilon}(\Lambda)$, i.e.
$\phi\in\ell^{p^{\prime}}(\Lambda)$ for all $p^{\prime}>p$, Lemma 5.1 implies
that $\delta(\phi)\leq\frac{1}{p^{\prime}}\delta$ and using Theorem 1 it
follows:

$\|\pi_{\phi}(f)\|_{\text{op}}\leq e^{\frac{p-1}{2p}\delta
r(f)}\|\lambda(f)\|_{\text{op}}$

The above estimate can be improved assuming $p\geq 2$ as follows:

Theorem 2. Let $(\pi,\mathcal{H}_{\pi})$ be a strongly $L^{p+\varepsilon}$
unitary representation with $2\leq p\neq\infty$, then:

$\|\pi(f)\|_{\text{op}}\leq e^{\frac{p-2}{2p}\delta
r(f)}\|\lambda(f)\|_{\text{op}}$

for all $f\in{\mathbb{C}}[\Lambda]$.

It would follows from the same arguments (or [31, Thm. 1]) that
$\|\pi_{\phi}(f)\|_{\text{op}}\leq\|\lambda(f)\|_{\text{op}}$ if
$\phi\in\ell^{p+\varepsilon}(\Lambda)\cap\mathbb{P}_{+}(\Lambda)$ for any
$1\leq p<2$. One the other hand Shalom’s trick [77, Lem. 2.3] shows that this
inequality is tight.

###### Proof.

We use the same notations as in Theorem 1 proof. Given ${p^{\prime}}>p$ we
consider
$\mathcal{H}_{\infty}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{D}_{p^{\prime}}\subset\mathcal{H}_{\pi}$
a dense subspace such that $(\pi(.)v,w)\in\ell^{p^{\prime}}(\Lambda)$ for all
$v,w\in\mathcal{D}_{p^{\prime}}$.

Note that for all $\kappa>0$, there exists $n_{0}\in{\mathbb{N}}$ such that

$|S^{(n)}|\leq|B(e,4nr(f))|\prec e^{(\delta+\kappa)4nr(f)}$

for all $n\geq n_{0}$. For $v\in\mathcal{H}_{\infty}$ and $\varepsilon$ such
that $p^{\prime}<p+\varepsilon$ one has:

$\displaystyle(\sum_{g\in\Gamma}(f^{*}*f)^{*2n}(g)(\pi_{\phi}(g)v,v))^{2}\leq\|(f^{*}*f)^{*2n}\|_{2}^{2}\sum_{S^{(n)}}|(\pi_{\phi}(g)v,v)|^{2}$
$\displaystyle\leq\|(f^{*}*f)^{*2n}\|_{2}^{2}|S^{(n)}|^{\frac{p+\varepsilon-2}{p+\varepsilon}}(\sum_{S^{(n)}}|(\pi_{\phi}(g)v,v)|^{p+\varepsilon})^{\frac{2}{p+\varepsilon}}$
$\displaystyle\prec(\sum_{g\in\Gamma}|(\pi_{\phi}(g)v,v)|^{p+\varepsilon})^{\frac{2}{p+\varepsilon}}\|(f^{*}*f)^{*2n}\|_{2}^{2}e^{\frac{p+\varepsilon-2}{p+\varepsilon}(\delta+\kappa)4nr(f)}$

and thus

$\displaystyle\|\pi_{\phi}(f)\|_{\text{op}}$
$\displaystyle=\sup_{v\in\mathcal{H}_{\infty}}\lim_{n}(\sum_{\Gamma}(f^{*}*f)^{*2n}(g)(\pi_{\phi}(g)v,v))^{\frac{1}{4n}}$
$\displaystyle\leq
e^{\frac{p+\varepsilon-2}{2p+2\varepsilon}(\delta+\kappa)r(f)}\|\lambda(f)\|_{\text{op}}$

As $\varepsilon,\kappa>0$ can be taken arbitrarily small:

$\|\pi_{\phi}(f)\|_{\text{op}}\leq e^{\frac{p-2}{2p}\delta
r(f)}\|\lambda(f)\|_{\text{op}}=e^{(s-\frac{1}{2})\delta
r(f)}\|\lambda(f)\|_{\text{op}}$

with $s=\frac{p-1}{p}$. ∎

######  Corollary 5.10.

Let $(\Lambda,d_{\Lambda})$ be a discrete metric group with the radial rapid
decay property and $(\beta_{r})_{r}$ be the uniform probabilities over the
balls of radius $r$. Then there exists $m\in{\mathbb{N}}$, for all
$\varepsilon>0$, there exists $C>0$, for all $\phi\in\mathbb{P}_{+}(\Lambda)$
:

$\|\pi_{\phi}(\beta_{r})\|_{\text{op}}\leq
Cr^{m}e^{\frac{1}{2}(\delta(\phi)+\varepsilon-\delta)r}$

In other words $\pi_{\phi}$ is exponentially mixing whenever
$\phi\in\mathbb{P}_{+}(\Lambda)$ has a critical gap, i.e
$\delta(\phi)<\delta$.

###### Proof.

Let $r_{0}\geq 0$ such that $|B(e,r)|\prec e^{(\delta+\varepsilon)r}$ for all
$r\geq r_{0}$. Using Theorem 1:

$\displaystyle\|\pi_{\phi}(\beta_{r})\|_{\text{op}}$ $\displaystyle\leq
e^{\frac{1}{2}(\delta(\phi))r}\|\lambda(\beta_{r})\|_{\text{op}}$
$\displaystyle\leq_{\text{RRD}}Cr^{m}e^{\frac{1}{2}(\delta(\phi))r}\|\beta_{r}\|_{2}$
$\displaystyle\leq
C^{\prime}r^{m}e^{\frac{1}{2}(\delta(\phi)+\varepsilon-\delta)r}$

∎

A stronger result can be deduced assuming $\Lambda$ has the RD property:

######  Corollary 5.11.

Let $(\Lambda,d_{\Lambda})$ be a discrete metric group with the rapid decay
property, $S\subset\Lambda$ a subset of critical exponent $\delta_{S}$ and
$(\beta_{r}(S))_{r}$ the uniform probabilities on $S\cap B(e,r)$. There
$m\in{\mathbb{N}}$, for all $\varepsilon>0$, there exists a constants $C>0$,
for all $\phi\in\mathbb{P}_{+}(\Lambda)$ :

$\|\pi_{\phi}(\beta_{r}(S))\|_{\text{op}}\leq
Cr^{m}e^{\frac{1}{2}(\delta(\phi)+\varepsilon-\delta_{S})r}$

######  Example 5.12.

Let $\Lambda$ with RD, $\phi\in\mathbb{P}_{+}(\Lambda)$ and $H\subset\Lambda$
such that $\delta(\phi)<\delta_{H}<\delta_{\Lambda}$ then $\pi_{\phi}|_{H}$ is
exponentially mixing. In particular if $H_{1},H_{2}\subset\Lambda$ and
$\delta_{H_{1}}<\delta_{H_{2}}<\delta_{\Lambda}$ then
$(\lambda|_{H_{2}},\ell^{2}(\Lambda/H_{1}))$ is exponentially mixing.

###### Corollary 5.11 proof.

Let $r_{0}\geq 0$ such that $|B(e,r)\cap S|\prec
e^{(\delta_{S}+\varepsilon)r}$ for all $r\geq r_{0}$. Using Theorem 1:

$\displaystyle\|\pi_{\phi}(\beta_{r})\|_{\text{op}}$ $\displaystyle\leq
e^{\frac{1}{2}(\delta(\phi))r}\|\lambda(\beta_{r}(S))\|_{\text{op}}$
$\displaystyle\leq_{\text{RD}}Cr^{m}e^{\frac{1}{2}(\delta(\phi))r}\|\beta_{r}(S)\|_{2}$
$\displaystyle\leq
C^{\prime}r^{m}e^{\frac{1}{2}(\delta(\phi)+\varepsilon-\delta_{S})r}$

∎

###  5.3. Quantitative property (T) for hyperbolic groups

This subsection is dedicated to the proof of Theorem 5.

Let $S\subset\Lambda$ be a finite subset of a discrete group $\Lambda$ and
$\varepsilon>0$. A isometric representation on a Hilbert space,
$(\pi,\mathcal{H}_{\pi})$, of $\Lambda$ has a $(S,\varepsilon)$-almost
invariant vector if there exists $v\in\mathcal{H}_{\pi}$ with $\|v\|=1$ such
that $\|\pi(g)v-v\|\leq\varepsilon$ for all $g\in S$. More generally a
representation $\pi$ has almost invariant vectors if $S\subset\Lambda$ finite
and $\varepsilon>0$ can be chosen arbitrarily.

######  Definition 5.13.

A discrete $\Lambda$ has the Kazhdan property (T) if every unitary
representation with almost invariant vectors has a non-zero invariant vector.

It follows from [82] that generic Gromov hyperbolic groups have the Kazhdan
property (T) (see [11] for more about property (T)).

######  Proposition 5.14.

Let $\Gamma$ be a Gromov hyperbolic group with the Kazhdan property (T). There
exists $0\leq\delta_{0}<\delta$ such that for all positive cyclic
representation without invariant vector, $\phi$, the critical exponent of
$\phi$ satisfies $\delta(\phi)\leq\delta_{0}$.

###### Proof.

It follows from [11, p.33] that if $\Gamma$ has the Kazhdan property (T) there
exist $S\subset\Gamma$ finite and $\varepsilon>0$ such that any representation
with a $(S,\varepsilon)$-almost invariant has a non-zero invariant vector.

Assume one can find $(\phi_{n}\simeq[\pi_{\phi_{n}},{\bf 1}_{\phi_{n}}])_{n}$
a sequence of positive cyclic representations without $(S,\varepsilon)$-almost
invariant vectors such that $\delta(\phi_{n})\rightarrow\delta$. It follows
from Lemma 4.3 and Theorem 8 that $\pi_{\phi_{n}}\xrightarrow{n}1$ for the
Fell topology. In particular there exists $N\in{\mathbb{N}}$, for all $n\geq
N$, $\pi_{\phi_{n}}$ has a $(S,\varepsilon)$-almost invariant vector which is
a contradiction. Therefore one can find $0\leq\delta_{0}<\delta$ such that
whenever $\phi\in\mathbb{P}_{+}(\Gamma)$ does not have a
$(S,\varepsilon)$-almost invariant vectors $\delta(\phi)\leq\delta_{0}$. ∎

######  Remark 5.15.

If we consider the spectrum of critical values:

$\text{Spec}_{+}(\Gamma,d_{\Gamma})\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\delta(\pi):[\pi,v]\in\mathbb{P}_{+}(\Gamma)\\}$

Proposition 5.14 implies that $1$ is isolated in
$\text{Spec}(\Gamma,d_{\Gamma})$.

###### Proof of Theorem 5.

Given a probability $\mu\in\text{Prob}(\Gamma)$ on $\Gamma$:

$\displaystyle\|\pi(\mu)v\|^{4}$
$\displaystyle=(\sum\mu^{*}*\mu(g)(\pi(g)v,v))^{2}$
$\displaystyle\leq\sum\mu^{*}*\mu(g)|(\pi(g)v,v)|^{2}$
$\displaystyle=\sum_{g}\mu^{*}*\mu(g)(\pi\otimes\overline{\pi}(g)v\otimes\overline{v},v\otimes\overline{v})$
$\displaystyle=\|\pi\otimes\overline{\pi}(\mu)v\otimes\overline{v}\|^{2}\leq\|\pi\otimes\overline{\pi}|_{W_{v}}(\mu)\|_{\text{op}}^{2}\|v\|^{4}$

with
$W_{v}=\text{Span}\\{\pi\otimes\overline{\pi}(g)v\otimes\overline{v}:g\in\Gamma\\}\subset\mathcal{H}\otimes\overline{\mathcal{H}}$.
As $[\pi\otimes\overline{\pi}|_{W_{v}},v\otimes\overline{v}]$ is a positive
cyclic representation and does not have invariant vectors since $\pi$ is
weakly mixing, it follows from Proposition 5.14 that
$\delta[\pi\otimes\overline{\pi}|_{W_{v}},v\otimes\overline{v}]\leq\delta_{0}$
for some $0\leq\delta_{0}<\delta$ that only depends on $(\Gamma,d_{\Gamma})$.
On the other hand Theorem 1 combined with rapid decay property for hyperbolic
groups [50] lead to

$\|\text{Avr}_{r}(\pi\otimes\pi|_{W_{v}})\|_{\text{op}}\leq
e^{\frac{1}{2}\delta_{0}r}\|\text{Avr}_{r}(\lambda)\|_{\text{op}}$

which implies

$\frac{\|\text{Avr}_{r}(\pi)v\|}{\|v\|}\leq\|\text{Avr}_{r}(\pi\otimes\pi|_{W_{v}})\|_{\text{op}}^{\frac{1}{2}}\leq
e^{\frac{1}{4}\delta_{0}r}\|\text{Avr}_{r}(\lambda)\|_{\text{op}}^{\frac{1}{2}}$

It follows that

$\|\text{Avr}_{r}(\pi)\|_{\text{op}}\leq
e^{\frac{1}{4}\delta_{0}r}\|\text{Avr}_{r}(\lambda)\|_{\text{op}}^{\frac{1}{2}}\leq
Cr(\mu)^{m}e^{\frac{1}{4}(\delta_{0}-\delta)r(\mu)}$

for $C>0$ and $m\in{\mathbb{N}}$ and thus
$h(\pi)\geq\frac{(\delta-\delta_{0})}{4}>0$. ∎

###  5.4. Spectral gap and boundary representations

We investigate the optimal mixing rate of $\phi\in\mathbb{P}_{+}(\Gamma)$ and
its boundary representations $\pi_{\phi}^{\partial}$.

We denote $\mu$ a quasi-invariant probability measure on $\overline{\Gamma}$
and $E$ a $(\Gamma,s,\mu)$-conditional expectation as in Definition 3.9.
Recall that $f\in L^{2}(\mu)\mapsto(E(f),f)_{L^{2}(\mu)}$ defines a
$\pi_{s}$-invariant quadratic form on $L^{2}(\mu)$.

######  Lemma 5.16.

Let $\Lambda$ be a discrete group, $(Z,\nu)$ a probability space such that
$\Lambda$ acts by measure class preserving transformations and
$f\in{\mathbb{R}}_{+}[\Lambda]$ a positive finitely supported function on
$\Lambda$. Let $B:\Lambda\times Z\rightarrow({\mathbb{R}},+)$ be a measurable
additive cocycle and $\pi_{B}$ the linear representation on $L^{2}(\nu)$ given
by $\pi_{B}(g)\psi=e^{-B(g;\bullet)}g^{*}\psi$ for $\psi\in L^{2}(\nu)$ and
$g\in\Lambda$. Then the bounded operator $\pi_{B}(f)$ on $L^{2}(\nu)$
satisfies

$\|\pi_{B}(f)|L^{2}(\nu)\|_{\text{op}}\leq\sqrt{\|\pi_{B}(f){\bf
1}|L^{\infty}(\nu)\|.\|\pi_{B}^{*}(f^{\vee}){\bf 1}|L^{\infty}(\nu)\|}$

where $f^{\vee}(g)=f(g^{-1})$ for $g\in\Lambda$.

###### Proof.

For $v,w\in L_{+}^{2}(\nu)$ one has:

$\displaystyle|(\pi(f)v,w)_{L^{2}(\nu)}|^{2}=|\int_{\overline{\Gamma}}\sum_{g\in\Gamma}f(g)e^{-B(g;z)}v(g^{-1}z)w(z)d\nu(z)|^{2}$
$\displaystyle\leq\int_{\overline{\Gamma}\times\Gamma}f(g)e^{-B(g;z)}|v|^{2}(g^{-1}z)d\nu(z)dg.\int_{\overline{\Gamma}\times\Gamma}f(g)e^{-B(g;z)}|w|^{2}(z)d\nu(z)dg$
$\displaystyle=(|v|^{2},\pi_{B}^{*}(f^{\vee}){\bf
1})_{L^{2}(\nu)}(\pi_{B}(f){\bf 1},|w|^{2})_{L^{2}(\nu)}$
$\displaystyle=\|\pi_{B}(f){\bf
1}|L^{\infty}(\nu)\|\|\pi_{B}^{*}(f^{\vee}){\bf
1}|L^{\infty}(\nu)\|\|v|L^{2}(\nu)\|^{2}\|w|L^{2}(\nu)\|^{2}$

which concludes the proof. ∎

######  Lemma 5.17.

Let $\mu$ be a quasi-invariant probability on $\overline{\Gamma}$,
$s\in[\frac{1}{2},1]$, $E$ a $(\Gamma,s,\mu)$-conditional expectation and
$f\in{\mathbb{R}}_{+}[\Gamma]$ with $f^{\vee}=f$. The bounded operator
$\pi_{s}(f)$ on $L^{2}(\mu)$ satisfies

$\|\pi_{s}(f)\|_{\text{op}}\leq\|\pi_{s}(f){\bf 1}|L^{\infty}(\mu)\|$

###### Proof.

As $E(\pi_{s}(g)\psi)=\pi_{s}^{*}(g)E(\psi)$ for $\psi\in L^{2}(\mu)$ and
$g\in\Gamma$ it follows $E(\pi_{s}(f){\bf 1})=\pi_{s}^{*}(f)E({\bf
1})=\pi_{s}^{*}(f){\bf 1}$ and thus

$\|\pi_{s}^{*}(f){\bf 1}\|_{\infty}=\|E(\pi_{s}(f){\bf
1})\|_{\infty}\leq\|\pi_{s}(f){\bf 1}\|_{\infty}$

as $E$ is a contraction on $L^{\infty}(\nu)$. The bound follows from the above
estimate and Lemma 5.16 with $(Z,\nu)=(\overline{\Gamma},\mu)$ and $B=sb$. ∎

Assuming $E$ is positive definite on $L^{2}(\mu)$, we denote
$\mathcal{H}(\mu,E)$ the Hilbert completion of $(L^{2}(\mu),E)$.

######  Corollary 5.18.

Let $\mu$ be a quasi-invariant probability on $\overline{\Gamma}$,
$s\in[\frac{1}{2},1]$, $E$ a positive definite $(\Gamma,s,\mu)$-conditional
expectation and $f\in{\mathbb{R}}_{+}[\Gamma]$ with $f^{\vee}=f$. The bounded
operator $\pi_{s}(f)$ on $\mathcal{H}(\mu,E)$ satisfies

$\|\pi_{s}(f)|\mathcal{H}(\mu,E)\|_{\text{op}}\leq\|\pi_{s}(f){\bf
1}|L^{\infty}(\mu)\|$

###### Proof.

As $\pi_{s}$ is unitary on $\mathcal{H}(\mu,E)$ and $f^{\vee}=f$, the operator
$\pi_{s}(f)$ is self-adjoint on $\mathcal{H}(\mu,E)$.

In particular $\pi_{s}(f)^{*}\pi_{s}(f)|_{L^{2}(\mu)}=\pi_{s}(f)^{2}$ and

$\displaystyle\|\pi_{s}(f)|\mathcal{H}(\mu,E)\|_{\text{op}}$
$\displaystyle\leq_{\text{Lem.
\ref{lem:tech:op:incl}}}\|\pi_{s}(f)^{*}\pi_{s}(f)|L^{2}(\mu)\|_{\text{op}}^{\frac{1}{2}}$
$\displaystyle\leq\|\pi_{s}(f)|L^{2}(\mu_{\sigma_{2}})\|_{\text{op}}$
$\displaystyle\leq_{\text{lem. \ref{lem:gener:cond}}}\|\pi_{s}(f){\bf
1}|L^{\infty}(\mu)\|$

∎

The following proposition gives a sharp norm estimates for certain radial
averages on $(\pi_{s},\mathcal{H}(\mu,E))$:

######  Proposition 5.19.

Let $\mu$ be a quasi-invariant probability on ${\partial}\Gamma$, $E$ a
positive definite $(\Gamma,s,\mu)$-conditional expectation. Then

$\|\sum_{g:d_{\Gamma}(g,e)\leq
L}\pi_{s}(g)|\mathcal{H}(\mu,E)\|_{\text{op}}\asymp\begin{cases}\frac{1}{(2s-1)}e^{s\delta
L}&\text{for $s>\frac{1}{2}$}\\\ e^{s\delta L}L&\text{for
$s=\frac{1}{2}$}\end{cases}$

for $L\geq R$.

###### Proof.

Observe that

$\displaystyle\inf_{{\partial}\Gamma}\pi_{s}(f){\bf 1}$
$\displaystyle\leq(E(\pi_{s}(f){\bf 1}),\pi_{s}(f){\bf
1})_{L^{2}(\mu)}^{\frac{1}{2}}\leq\|\pi_{s}(f)|\mathcal{H}(\mu,E)\|_{\text{op}}$
$\displaystyle\leq_{\text{Cor. \ref{cor:control:general:l2}}}\|\pi_{s}(f){\bf
1}|L^{\infty}(\mu)\|$

and thus $\|\pi_{s}(f)|\mathcal{H}(\mu,E)\|_{\text{op}}\asymp\|\pi_{s}(f){\bf
1}|L^{\infty}(\mu)\|$ whenever $\|\pi_{s}(f){\bf
1}|L^{\infty}(\mu)\|\prec\inf_{{\partial}\Gamma}\pi_{s}(f){\bf 1}$.

We conclude using Lemma 2.14. ∎

######  Corollary 5.20.

Let $\mu$ be a quasi-invariant probability on ${\partial}{\Gamma}$,
$s\in[\frac{1}{2},1]$ and $E$ a positive definite $(\Gamma,s,\mu)$-conditional
expectation. Then

$\lim_{r\rightarrow+\infty}-\frac{1}{r}\log\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{s}(g)|\mathcal{H}(\mu,E)\|_{\text{op}}=\delta-s\delta$

###### Proof.

It follows from Proposition 5.19 and pure sphericality of Gromov hyperbolic
groups [29, Thm. 7.2] that

$\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
L}\pi_{s}(g)|\mathcal{H}(\mu,E)\|_{\text{op}}\asymp\begin{cases}\frac{1}{(2s-1)}e^{-(1-s)\delta
L}&\text{for $s>\frac{1}{2}$}\\\ e^{-\frac{1}{2}\delta L}L&\text{for
$s=\frac{1}{2}$}\end{cases}$

and thus

$\lim_{r\rightarrow+\infty}-\frac{1}{r}\log\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{s}(g)|\mathcal{H}(\nu,E)\|_{\text{op}}=(1-s)\delta$

∎

In particular the boundary representations, $\pi_{\phi}^{\partial}$, are
exponentially mixing with entropy equal to
$\delta-\max\\{\delta(\phi),\frac{1}{2}\delta\\}$.

######  Remark 5.21.

Given $\phi,\phi^{\prime}\in\mathbb{P}_{+}(\Gamma)$ with
$\delta(\phi),\delta(\phi^{\prime})\geq\frac{1}{2}\delta$ and their respective
boundary representations $\pi_{\phi}^{\partial}$ and
$\pi_{\phi^{\prime}}^{\partial}$, a necessary condition for these
representations to be unitary equivalent is that
$\delta(\phi)=\delta(\phi^{\prime})$.

######  Corollary 5.22.

Given $\phi\in\mathbb{P}_{+}(\Gamma)$ one has:

$\lim_{r\rightarrow+\infty}-\frac{1}{r}\log\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{\phi}(g)\|_{\text{op}}=\delta-\max\\{\delta(\phi),\frac{1}{2}\delta\\}$

###### Proof.

Let $\sigma\delta>\max\\{\delta(\phi),\frac{1}{2}\delta\\}$ and $J_{\sigma}$
the intertwiner considered in Subsection 2.5 with
$\omega_{\sigma}(g)=e^{-\sigma\delta d_{\Gamma}(g,e)}$ for $g\in\Gamma$ (see
also Subsection 3.2). One has:

$\displaystyle\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq r}\pi_{\phi}(g)$
$\displaystyle|\mathcal{H}_{\phi}\|_{\text{op}}=\|J_{\sigma}(\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{\sigma}(g))J_{\sigma}^{-1}|\mathcal{H}_{\phi}\|_{\text{op}}$
$\displaystyle=\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{\sigma}(g)|\mathcal{H}(\mu_{\sigma},E_{\sigma})\|_{\text{op}}$
$\displaystyle\leq_{Cor.\ref{cor:control:general:l2}}\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{\sigma}(g){\bf 1}\|_{\infty}$ $\displaystyle\prec_{\text{Cor.
\ref{cor:ball:spec:tech}}}e^{-(1-\sigma)\delta r}$

It follows that

$\delta-\sigma\delta\leq\liminf_{r}-\frac{1}{r}\log\|\frac{1}{|B(e,r)|}\sum_{g:d_{\Gamma}(g,e)\leq
r}\pi_{\phi}(g)|\mathcal{H}_{\phi}\|_{\text{op}}$

for all $\sigma>\max\\{s(\phi),\frac{1}{2}\\}$. On the other hand Lemma 4.3
implies that $\pi_{\phi}^{\partial}\prec\pi_{\phi}$ and thus

$\limsup_{r}-\frac{1}{r}\log\|\pi_{\phi}(\beta_{r})\|\leq\limsup_{r}-\frac{1}{r}\log\|\pi_{\phi}^{\partial}(\beta_{r})\|\leq_{\text{lem
\ref{cor:bound:rep:mixing}}}\delta-\max\\{\delta(\phi),\frac{1}{2}\delta\\}$

∎

In particular:

######  Corollary 5.23.