# Uniform resonance free regions for convex cocompact hyperbolic surfaces and
expanders

Louis Soares<EMAIL_ADDRESS>
###### Abstract.

Let $X=\Gamma\backslash\mathbb{H}^{2}$ be a convex cocompact hyperbolic
surface, let $X_{n}=\Gamma_{n}\backslash\mathbb{H}^{2}$ be a family of finite
regular covers of $X$, and let $\mathcal{G}_{n}$ be the associated family of
Cayley graphs of the covering groups $G_{n}=\Gamma/\Gamma_{n}$ with respect to
the set of Schottky generators of $X$. We prove that if the graphs
$\mathcal{G}_{n}$ form a family of two-sided expanders, then the surfaces
$X_{n}$ have a uniform resonance-free strip. This is the analogue for convex
cocompact hyperbolic surfaces of the results of Brooks [10], Burger [11, 12],
and Bourgain–Gamburd–Sarnak [6]. Moreover, this result generalizes the work on
convex cocompact congruence covers by Oh–Winter [32] and
Bourgain–Kontorovich–Magee [20].

###### Key words and phrases:

Resonances, hyperbolic surfaces, expanders, Selberg zeta function, transfer
operators

###### 2010 Mathematics Subject Classification:

Primary: 58J50, 11M36, Secondary: 37C30, 37D35, 05C90

##  1\. Introduction

###  1.1. Spectral gap and expanders

The relation between spectral gaps of the Laplacian on compact Riemannian
manifolds and expander graphs is well-known in the literature, mainly through
the work of Brooks [10] and Burger [11, 12].

Expanders are highly-connected sparse graphs with many applications in
computer science and pure mathematics. There is a vast literature on expanders
and we refer the reader to [18, 26, 39, 24] for a comprehensive discussion.
Expanders are usually defined in terms of the Cheeger (isoperimetric)
constant. For an undirected graph $\mathcal{G}$ with vertex set $V$ and a
subset $A\subset V$, the boundary $\partial A$ of $A$ is defined as the set of
edges with one extremity in $A$ and the other in $V\smallsetminus A$. The
Cheeger constant of $\mathcal{G}$ is defined by

$h(\mathcal{G})\stackrel{{\scriptstyle\text{def}}}{{=}}\min\left\\{\frac{|\partial
A|}{|A|}:A\subset V,\,1\leq|A|\leq\frac{|V|}{2}\right\\},$

and a family of $k$-regular graphs $\mathcal{G}_{n}$ is said to form a family
of expanders if $h(\mathcal{G}_{n})$ is bounded below by some uniform positive
constant.

Given a compact Riemannian manifold $X$, a family of finite coverings $X_{n}$
of $X$ and a set $S$ of generators of the fundamental group $\pi_{1}(X)$, we
can naturally build a family of graphs $\mathcal{G}_{n}$ as follows. Note that
$\pi_{1}(X_{n})$ can be viewed as a subgroup of $\pi_{1}(X)$ for each $n$, so
we define the vertices of $\mathcal{G}_{n}$ to be the cosets of
$\pi_{1}(X)/\pi_{1}(X_{n})$ (of which there are finitely many) and we join two
vertices of $\mathcal{G}_{n}$ by an edge if and only if the corresponding
cosets differ by left multiplication by an element in $S$. If we assume
furthermore that the $X_{n}$’s are regular coverings of $X$, then we can view
each $\pi_{1}(X_{n})$ as a normal subgroup of $\pi_{1}(X)$. In this case the
quotient $G_{n}=\pi_{1}(X)/\pi_{1}(X_{n})$ forms a group and the graphs
$\mathcal{G}_{n}$ are the Cayley graphs of the group $G_{n}$ with respect to
the set $S_{n}\subset G_{n}$, where $S_{n}$ is the image of the set $S$ of
generators under the natural projection map $\pi_{1}(X)\to G_{n}$.

The aforementioned result of Brooks [10] and Burger [11, 12] says that the
Laplacians acting on the $X_{n}$’s have a uniform positive spectral gap if and
only if the graphs $\mathcal{G}_{n}$ form a family of expanders.
Bourgain–Gamburd–Sarnak [6] extended this result to the case where $X$ is a
geometrically finite hyperbolic surface with cusps. In the aforementioned
papers, the $X_{n}$’s are said to have uniform spectral gap if there exists
some constant $c>0$ such that

$\lambda_{1}(X_{n})\geq\lambda_{0}(X_{n})+c,$

holds true for each $n$, where

$0\leq\lambda_{0}(X_{n})<\lambda_{1}(X_{n})\leq\cdots$

is the sequence of $L^{2}$-eigenvalues of the positive Laplacian on $X_{n}$.
In the present paper, we focus on the case where
$X=\Gamma\backslash\mathbb{H}^{2}$ is a convex cocompact hyperbolic surface,
that is, a non-compact hyperbolic surface with a finite number of funnel ends
and no cusps (see [3]).

To properly state our result, we briefly review some aspects of the spectral
theory of infinite-area hyperbolic surfaces, referring the reader to
Borthwick’s book [3]. We assume that $\Gamma$ is a finitely generated, non-
cofinite Fuchsian group and we let $X=\Gamma\backslash\mathbb{H}^{2}$ be the
associated hyperbolic surface. The limit set $\Lambda_{\Gamma}$ is defined as
the set of accumulation points of orbits of $\Gamma$ acting on the hyperbolic
plane $\mathbb{H}^{2}$. It is a subset of the boundary of $\mathbb{H}^{2}$ at
infinity, the latter of which we identify with
$\overline{\mathbb{R}}=\mathbb{R}\cup\\{\infty\\}$. The limit set is a Cantor-
like fractal subset and its Hausdorff dimension, which we denote by
$\delta_{\Gamma}$, coincides with the exponent of convergence of Poincaré
series for $\Gamma$.

The $L^{2}$-spectrum of the positive Laplacian $\Delta_{X}$ on $X$ has been
described completely by Lax–Phillips [25]. The continuous spectrum of
$\Delta_{X}$ coincides with the interval $[1/4,\infty)$, which does not
contain any embedded $L^{2}$-eigenvalues. The pure point spectrum is empty if
$\delta_{\Gamma}\leq\frac{1}{2}$ and finite and starting at
$\delta_{\Gamma}(1-\delta_{\Gamma})$ if $\delta_{\Gamma}>\frac{1}{2}$. This
description of the spectrum shows that the resolvent

$R_{X}(s)=\left(\Delta_{X}-s(1-s)\right)^{-1}:L^{2}(X)\rightarrow L^{2}(X)$

is well-defined and analytic on the half-plane $\mathrm{Re}(s)>\frac{1}{2}$
except at the finite set of poles corresponding to the pure point spectrum of
$\Delta_{X}$. The resonances of $X$ are the poles of the meromorphic
continuation of

(1) $R_{X}(s):C_{0}^{\infty}(X)\rightarrow C^{\infty}(X)$

to the whole complex plane. This continuation can be deduced from the analytic
Fredholm theorem using an adequate parametrix provided by Guillopé–Zworski
[17]. In the sequel, we denote by ${\mathcal{R}}_{X}$ the set of resonances of
$X$. The resolvent operator (1) has a simple pole at $s=\delta_{\Gamma}$ and
no other poles in the half-plane $\mathrm{Re}(s)\geq\delta_{\Gamma}$, by a
result of Patterson [34]. In other words, $s=\delta_{\Gamma}$ is the resonance
of $X$ with the largest real part.

We also recall that by Patterson–Perry [35], resonances appear as zeros of the
Selberg zeta function, which is defined by

(2)
$Z_{\Gamma}(s)\stackrel{{\scriptstyle\text{def}}}{{=}}\prod_{k=0}^{\infty}\prod_{[\gamma]\in[\Gamma]_{\mathsf{prim}}}\left(1-e^{-(s+k)\ell(\gamma)}\right)$

for $\mathrm{Re}(s)\gg 1.$ The inner product in (2) is taken over the set
$[\Gamma]_{\mathsf{prim}}$ of conjugacy classes of primitive hyperbolic
elements of $\Gamma$.

###  1.2. Statement of main theorem

Given a finite-index normal subgroup $\Gamma^{\prime}\lhd\Gamma,$ the limit
sets $\Lambda_{\Gamma^{\prime}}$ and $\Lambda_{\Gamma}$ coincide and in
particular, so do their Hausforff dimensions. Therefore $s=\delta_{\Gamma}$ is
a common simple zero of $Z_{\Gamma}(s)$ and $Z_{\Gamma^{\prime}}(s),$ and a
common simple resonance of
$X^{\prime}=\Gamma^{\prime}\backslash\mathbb{H}^{2}$ and
$X=\Gamma\backslash\mathbb{H}^{2}.$ More generally, in light of the
Venkov–Zograf factorization formula (22), every zero of $Z_{\Gamma}(s)$ is
also a zero of $Z_{\Gamma^{\prime}}(s)$. Thus, we introduce the following
terminology: we say that $s$ is a new zero of $Z_{\Gamma^{\prime}}(s)$ with
respect to $\Gamma$ if it is a zero of $Z_{\Gamma^{\prime}}(s)$ but not a zero
of $Z_{\Gamma}(s).$ Likewise, we say that $s$ is a new resonance of
$X^{\prime}$ with respect to $X$ if it belongs to $\mathcal{R}_{X}$ but not to
$\mathcal{R}_{X^{\prime}}.$

By Button’s work [13] every convex cocompact hyperbolic surface $X$ is
isometric to a quotient $\Gamma\backslash\mathbb{H}^{2}$ where $\Gamma$ is a
Schottky group. Schottky groups stand out, among other Fuchsian groups, by
their simple geometric construction. For the rest of this introduction, we fix

a Schottky group $\Gamma$ with Schottky generators
$\gamma_{1},\dots,\gamma_{m}$ with $m\geq 2$

(see Section 2.1 for definitions and properties). We let
$\Gamma_{n}\lhd\Gamma$ be a family of finite-index normal subgroups of
$\Gamma$ and we let $X_{n}=\Gamma_{n}\backslash\mathbb{H}^{2}$ be the
associated family of regular covers of $X.$ Moreover, we let
$\mathcal{G}_{n}=\mathrm{Cay}(G_{n},\pi_{n}(S))$ be the Cayley graphs of the
finite groups $G_{n}=\Gamma/\Gamma_{n}$ with respect to the image in $G_{n}$
of the generators $S=\\{\gamma_{1}^{\pm},\dots,\gamma_{m}^{\pm}\\}$ under the
natural projection map $\pi_{n}\colon\Gamma\to G_{n}.$

######  Theorem 1.1 (Main theorem).

Let assumptions and notations be as above. Assume furthermore that for every
$n,$ the map $\pi_{n}$ is injective on the set
$S=\\{\gamma_{1}^{\pm},\dots,\gamma_{m}^{\pm}\\}$ to ensure that
$\mathcal{G}_{n}$ is $2m$-regular. Then, for every $\epsilon>0$ there exists
$\eta=\eta(\epsilon)>0$ such that if the graphs $\mathcal{G}_{n}$ are two-
sided $\epsilon$-expanders, the Selberg zeta function $Z_{\Gamma_{n}}(s)$ has
no new zeros with respect to $\Gamma$ in the half-plane

$\mathrm{Re}(s)\geq\delta_{\Gamma}-\eta.$

Equivalently, if $\mathcal{G}_{n}$’s form a family of are two-sided
$\epsilon$-expanders, then the surfaces
$X_{n}=\Gamma_{n}\backslash\mathbb{H}^{2}$ have a uniform relative resonance
gap of size $\eta,$ that is

(3)
$\mathcal{R}_{X_{n}}\cap\\{\mathrm{Re}(s)\geq\delta_{\Gamma}-\eta\\}=\mathcal{R}_{X}\cap\\{\mathrm{Re}(s)\geq\delta_{\Gamma}-\eta\\}.$

We will give a precise definition of the notion two-sided $\epsilon$-expanders
further below. Let us first give a couple of remarks.

  1. (1)

By Naud [27] we know that each hyperbolic surface $X$ has an individual
resonance-free strip, that is, there exists $\eta>0$ such that, other than the
trivial resonance at $s=\delta_{\Gamma},$ there are no resonances in the half-
plane $\mathrm{Re}(s)>\delta_{\Gamma}-\eta.$ (See also the more recent work of
Bourgain–Dyatlov [4].)

  2. (2)

For elementary Schottky groups $\Gamma=\langle\gamma\rangle$ (that is, when
$m=1$) the associated quotient $X=\mathbb{H}^{2}/\langle\gamma\rangle$ is a
hyperbolic cylinder. In this case, the resonances of $X$ are given explicitly
by $\mathcal{R}_{X}=(2\pi i\ell)\mathbb{Z}-\mathbb{N}_{0}$ where $\ell>0$ is
the displacement length of $\gamma$ (see Proosition 5.1 in [3]). It is clear
from this formula that Theorem 1.1 fails when $\Gamma$ is elementary.

  3. (3)

For $\delta_{\Gamma}>\frac{1}{2}$ the statement in Theorem 1.1 follows from
Theorem 1.2 in [6] because every resonance $s\in\mathcal{R}_{X}$ with
$\mathrm{Re}(s)>\frac{1}{2}$ corresponds to an $L^{2}$-eigenvalue
$\lambda=s(1-s)$ of the Laplacian on $X$. On the other hand, by the results of
Beardon [1, 2], we know that $\delta_{\Gamma}\leq\frac{1}{2}$ implies that
$\Gamma$ has no cusps. Thus the case $\delta_{\Gamma}\leq\frac{1}{2}$ comes
under the purview of Theorem 1.1.

  4. (4)

We note that the methods used by Brooks [10], Burger [11, 12] and
Bourgain–Gamburd–Sarnak [6] rely solely on $L^{2}$-methods, which are no
longer available in the range $\delta_{\Gamma}\leq\frac{1}{2}.$

  5. (5)

We conjecture the following converse of Theorem 1.1: if (3) holds true for the
surfaces $X_{n}$, then the graphs $\mathcal{G}_{n}$ form a family of
expanders.

While expanders are usually defined in terms of the Cheeger constant, it will
be more convenient for us to use the spectral notion of expansion. Let $G$ be
a finite group, let $\epsilon>0$ and let $g_{1},\dots,g_{m}\in G$. On the
Hilbert space $L^{2}(G)$ of functions $f\colon G\to\mathbb{C}$ with norm

$\|f\|_{L^{2}(G)}\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\sum_{x\in
G}|f(x)|^{2}\right)^{1/2},$

we define the operator $T\colon L^{2}(G)\to L^{2}(G)$ by

$Tf(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{2m}\sum_{j=1}^{m}\left(f(xg_{j})+f(xg_{j}^{-1})\right).$

for all $x\in G$ and all $f\in L^{2}(G).$ We say that the set
$\\{g_{1},\dots,g_{m}\\}$ is $\epsilon$-expanding if one has

$\|Tf\|_{L^{2}(G)}\leq(1-\epsilon)\|f\|_{L^{2}(G)}$

for all functions $f\colon G\to\mathbb{C}$ of mean zero. Equivalently, we say
$\\{g_{1},\dots,g_{m}\\}$ is $\epsilon$-expanding if all the eigenvalues of
the self-adjoint operator $T$, other than the trivial eigenvalue $1$, lie in
the interval $[-1+\epsilon,1-\epsilon].$

A $d$-regular graph $\mathcal{G}$ is said to be a two-sided
$\epsilon$-expander if the eigenvalues of the Laplacian on $\mathcal{G},$
other than the trivial eigenvalue $d$, lie in the interval
$[d(-1+\epsilon),d(1-\epsilon)].$ This notion of expansion is closely related
to but slightly stronger than the definition of expanders in terms of the
Cheeger constant. If $\mathcal{G}$ is a two-sided $\epsilon$-expander, then it
is also an expander in the sense that its Cheeger constant is bounded below by
some constant depending on $\epsilon.$ Observe furthermore that the Cayley
graph

$\mathcal{G}=\mathrm{Cay}(G,\\{g_{1}^{\pm},\dots,g_{m}^{\pm}\\})$

is a two-sided $\epsilon$-expander if and only if the set
$\\{g_{1},\dots,g_{m}\\}$ is $\epsilon$-expanding. We can therefore give an
alternative and more convenient111In fact, Theorem 1.2 is slightly stronger in
that we can drop the injectivity assumption from the statement formulation of
the main theorem without having to resort to graphs:

######  Theorem 1.2 (Main theorem, reformulated).

For every $\epsilon>0$ there exists $\eta=\eta(\epsilon)>0$ such that the
following statement holds true. Let $G$ be a finite group and let
$\Gamma^{\prime}$ be the kernel of a homomorphism $\pi\colon\Gamma\to G$. If
the set $\\{\pi(\gamma_{1}),\dots,\pi(\gamma_{m})\\}\subseteq G$ is
$\epsilon$-expanding, then $Z_{\Gamma^{\prime}}(s)$ has no new zeros with
respect to $\Gamma$ in the half-plane $\mathrm{Re}(s)\geq\delta-\eta.$

Our proof method will give a dependence of $\eta$ on $\epsilon$ in Theorems
1.1 and 1.2 of the form

(4) $\eta(\epsilon)=c_{1}\exp\left(-c_{2}\epsilon^{-1}\right),$

where $c_{1},c_{2}$ are positive constants depending only on the set of
Schottky generators of $\Gamma.$ We have made no attempt to effectively
calculate these constants, although such a calculation would be feasible at
the expense of a more technical proof. The dependence on $\epsilon$ in (4) is
probably far from being optimal and we conjecture that the same conclusion
holds true with $\eta(\epsilon)=c_{1}\epsilon^{c_{2}}$ for some suitable
constants $c_{1}$ and $c_{2}.$

Theorem 1.2 may be used to establish a uniform resonance-free strip for the
family of congruence covers. When $\Gamma$ a subgroup in
$\mathrm{SL}_{2}(\mathbb{Z})$ and $q\in\mathbb{N}$, the congruence subgroup of
$\Gamma$ of level $q$ is defined by

$\Gamma(q)\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{\text{$\gamma\in\Gamma$
: $\gamma\equiv I$ mod $q$}\right\\},$

or equivalently, $\Gamma(q)$ is the kernel of the reduction map modulo $q$,

$\pi_{q}\colon\Gamma\to\mathrm{SL}_{2}(\mathbb{Z}/q\mathbb{Z}),\quad\gamma\mapsto\text{$\gamma$
mod $q$.}$

The congruence cover of $X$ of level $q$ is the quotient
$X(q)=\Gamma(q)\backslash\mathbb{H}^{2}.$ When $X$ is convex cocompact, the
existence of a uniform resonance-free strip for the surfaces $X(q)$ was first
proven by Oh–Winter [32] for $q$ running through the positive square-free
integers, and later extended by Bourgain–Kontorovich–Magee [20] to all
$q\in\mathbb{N}$ without small divisors. In both papers, one of the main
ingredients is the deep result of Bourgain–Varjú [7], which implies that there
exists a constant $\epsilon=\epsilon(\Gamma)>0$ such that for all
$q\in\mathbb{N},$ the set
$\\{\pi_{q}(\gamma_{1}),\dots,\pi_{q}(\gamma_{m})\\}$ is $\epsilon$-expanding
in the group

$G_{q}\stackrel{{\scriptstyle\text{def}}}{{=}}\pi_{q}(\Gamma)\leqslant\mathrm{SL}_{2}(\mathbb{Z}/q\mathbb{Z}).$

When combined with Theorem 1.2 above, this immediately implies the following:

######  Theorem 1.3 (Resonance-free strip for $X(q)$ for arbitrary $q$).

Let $\Gamma\subset\mathrm{SL}_{2}(\mathbb{Z})$ be a convex cocompact Fuchsian
group. There exists $\eta>0$ depending only on $\Gamma$ such for all positive
integers $q$, the Selberg zeta function $Z_{\Gamma(q)}(s)$ has no new zeros
with respect to $\Gamma$ in the half-plane
$\mathrm{Re}(s)\geq\delta_{\Gamma}-\eta$. Equivalently, the surfaces $X(q)$
have no new resonances with respect to $X$ in
$\mathrm{Re}(s)\geq\delta_{\Gamma}-\eta.$

Actually, this theorem is true for all non-elementary, finitely generated
Fuchsian groups $\Gamma\subset\mathrm{SL}_{2}(\mathbb{Z})$. For the modular
group $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$ (in which case we have
$\delta_{\Gamma}=1$) Selberg’s famous $\frac{3}{16}$-theorem [40] gives a more
precise statement with the explicit spectral gap $\eta=\frac{1}{4}$. For
$\delta_{\Gamma}>\frac{5}{6}$, Theorem 1.3 follows from Gamburd’s thesis [15]
with $\eta=\delta_{\Gamma}-\frac{5}{6}$. For $\delta_{\Gamma}>\frac{1}{2}$,
the statement was proven by Bourgain–Gamburd–Sarnak [6]. The more difficult
case $\delta_{\Gamma}\leq\frac{1}{2}$ is covered by Theorem 1.3 (recall that
$\delta_{\Gamma}>\frac{1}{2}$ implies that $\Gamma$ has at least one cusp by
the results of Beardon [1, 2]). In view of the arithmetic applications of
Theorem 1.3 (for which we refer to the survey paper of Kontorovich [22]), it
would be interesting to have explicit bounds for $\eta$. Selberg’s conjecture
suggests that $\eta=\delta/2$ is the optimal gap, which seems out of reach at
the current state of the art. Good bounds for $\eta$ in Theorem 1.3 are
difficult to obtain, due to the reliance on the expander theory developed
among others in Bourgain–Varju [7] and Bourgain–Gamburd [5]. These papers do
not provide explicit bounds for explicit expansion coefficients for the Cayley
graphs
$\mathcal{G}_{q}=\mathrm{Cay}(G_{q},\\{\pi_{q}(\gamma_{1})^{\pm},\dots,\pi_{m}(\gamma_{1})^{\pm}\\})$.
The first explicit bounds for the spectral gap of the Laplacian on
$\mathcal{G}_{q}$ when $q$ is prime was obtained by Kowalski [23]. However,
inserting these bounds into (4) would yield values for $\eta$ which are very
unlikely to be anywhere near sharp.

###  1.3. Notation

We write $f(x)\ll g(x)$ or $f(x)=O(g(x))$ interchangeably to mean that there
exists an implied constant $C>0$ such that $|f(x)|\leq C|g(x)|$ for all $x\geq
C.$ Whenever the Schottky group $\Gamma$ is fixed we will drop it from the
subscripts, writing $\delta=\delta_{\Gamma}$ and $\Lambda=\Lambda_{\Gamma}.$
We write $C=C(\Gamma,x,y,\dots)$ to emphasize that the constant $C$ depends on
the Schottky data of $\Gamma$ and on the variables $x,y,\dots$. For a finite
set $A$ we denote by $|A|$ its cardinality and we write

$\mathbb{E}_{a\in
A}f(a)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{|A|}\sum_{a\in A}f(a).$

For a positive integer $N$ we use the notation $[N]=\\{1,\dots,N\\}.$

###  1.4. Outline of the proof

We now give an outline of the proof of Theorem 1.2, glossing over all the
technical details. Suppose by contradiction that $Z_{\Gamma^{\prime}}(s)$ has
a new zero at $s=\sigma+it$ with $\sigma>\delta-\eta,$ where
$\eta=\eta(\epsilon)>0$ is chosen at a later stage in the proof. By the
Venkov–Zograf formula, $s$ is a zero of a twisted Selberg zeta function
$Z_{\Gamma}(s,\rho),$ where $\rho\colon G\to\mathrm{U}(V)$ is a a non-trivial,
irreducible representation of the group $G$. By the thermodynamical formalism,
zeros of $Z_{\Gamma}(s,\rho)$ correspond to 1-eigenfunctions of a transfer
operator $\mathcal{L}_{s,\rho}$ acting on spaces of $V$-valued functions on
the Schottky disks of $\Gamma$. That is, there exists a non-zero function $F$
such that

(5) $\mathcal{L}_{s,\rho}^{N}F=F$

for all $N\in\mathbb{N}.$ By normalizing equation (5), this is equivalent to
the existence of functions $f$ on the limit set $\Lambda$ such that

(6) $\widetilde{\mathcal{L}}_{s,\rho}^{N}f=e^{-NP(\sigma)}f,$

where $\widetilde{\mathcal{L}}_{s,\rho}$ is a normalized version of the
operator $\mathcal{L}_{s,\rho}$, and $\sigma\mapsto P(\sigma)$ is the
topological pressure of the Bowen–Series map that encodes the group $\Gamma.$
The goal now is to show that non-zero functions $f$ satisfying (6) can not
exist whenever $\sigma=\mathrm{Re}(s)$ is sufficiently close to $\delta.$

To proceed, we recall that $\Gamma$ is generated by the elements
$\gamma_{1},\dots,\gamma_{m}$ and their inverses, which we denote by

$\gamma_{m+1}=\gamma_{1}^{-1}$, $\gamma_{m+2}=\gamma_{2}^{-1},$ … ,
$\gamma_{2m}=\gamma_{m}^{-1}$.

We also note that the elements of $\Gamma$ are indexed by reduced words in the
alphabet $[2m]=\\{1,\dots,2m\\}$. Given a word $\textbf{a}=a_{1}\cdots a_{N}$
with letters $a_{1},\dots,a_{N}$ in $[2m]$, we write

$\gamma_{\textbf{a}}\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{a_{1}}\circ\cdots\circ\gamma_{a_{N}}.$

We let $W_{N}^{j}$ denote the set of reduced words of length $N$ not ending
with the letter $j.$ Then, by Proposition 4.1, if $x\in\Lambda$ is a maximizer
of the function $x\mapsto\|f(x)\|_{V},$ there exist an index $j\in[2m]$ such
that

$f(x)=\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))+\Delta_{\textbf{a},N}$

for all words $\textbf{a}\in W_{N}^{j},$ where $\Delta_{\textbf{a},N}$ is an
element of $V$ whose norm is small when $\sigma$ is close to $\delta.$ In
fact, for any fixed $N$ and $\textbf{a},$ we have
$\|\Delta_{\textbf{a},N}\|_{V}\to 0$ as $\sigma\to\delta^{-}.$ For the sake of
exposition, we assume that $N$ and $\eta$ have already been chosen such that
for all $\sigma\geq\delta-\eta,$ the norm of $\Delta_{\textbf{a},N}$ is small
enough for our purposes, and we will simply write

(7)
$f(x)\approx\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x)).$

We could now average both sides of (7) over all words $\textbf{a}\in
W_{N}^{j}$ to obtain

$f(x)\approx\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right).$

For $t=0$ we could use the key estimate in Proposition 3.3 (which is where we
take advantage of the expansion property) to estimate the average on the left
hand side, which gives

$\left\|\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)\right\|_{V}\ll
e^{-c\epsilon N}\|f\|_{\infty,\Lambda},$

where $\|f\|_{\infty,\Lambda}$ is the maximum norm of $f$ on the limit set.
This would give a contradiction when $N$ is sufficiently large. The proof of
this estimate relies on a decoupling argument similar to the one contained in
[6]. For $t\neq 0$, however, the factors
$\gamma_{\textbf{a}}^{\prime}(x)^{it}$ interfere with this argument in a
rather inconvenient way. To overcome this issue, we take the tensor product of
both sides of (7) with its complex conjugate, which gives

$\displaystyle f(x)\otimes\overline{f(x)}$
$\displaystyle\approx\left(\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)\otimes\overline{\left(\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)}$
$\displaystyle=\left(\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)\otimes\left(\gamma_{\textbf{a}}^{\prime}(x)^{-it}\overline{\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))}\right)$
$\displaystyle=\left(\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)\otimes\overline{\left(\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)}.$

By defining

$h\stackrel{{\scriptstyle\text{def}}}{{=}}f\otimes\overline{f}$ and
$\nu\stackrel{{\scriptstyle\text{def}}}{{=}}\rho\otimes\overline{\rho},$

the above takes the form

(8) $h(x)\approx\nu(\gamma_{\textbf{a}})^{-1}h(\gamma_{\textbf{a}}(x)).$

Note that the complex factors $\gamma_{\textbf{a}}^{\prime}(x)^{it}$ are no
longer present here, so we can now average (8) over $\textbf{a}\in W_{N}^{j}$
to get

(9) $h(x)\approx\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h(\gamma_{\textbf{a}}(x))\right),$

and use the key estimate in Proposition 3.3 to estimate the right hand side of
(9). Since the function $h$ might not satisfy the necessary assumptions, we
must first decompose it as

$h=h_{1}+h_{2},$

where $h_{1}$ is the projection onto the “trivial” subspace
$W_{\mathrm{triv}}\subseteq W=V\otimes V.$ We can now apply the estimate to
$h_{2}=h-h_{1}$ (which satisfies the required assumptions) to obtain

(10) $\left\|\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h_{2}(\gamma_{\textbf{a}}(x))\right)\right\|_{W}\ll
e^{-c\epsilon N}\|h\|_{\infty,\Lambda}.$

On the other hand, as an application of Schur’s lemma, we have

(11)
$\|h_{1}\|_{\infty,\Lambda}\leq\frac{1}{\sqrt{\dim(\rho)}}\|h\|_{\infty,\Lambda}.$

Going back to (9) and using the triangle inequality, we can show that

$\|h\|_{\infty,\Lambda}\lesssim\|h_{1}\|_{\infty,\Lambda}+\left\|\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h_{2}(\gamma_{\textbf{a}}(x))\right)\right\|_{W}.$

Invoking the estimates (10) and (11) gives

$\|h\|_{\infty,\Lambda}\leq\left(\frac{1}{\sqrt{\dim(\rho)}}+O(e^{-c\epsilon
N})\right)\|h\|_{\infty,\Lambda}.$

Assuming that the dimension of $\rho$ is at least two, we obtain furthermore

$\|h\|_{\infty,\Lambda}\leq\left(\frac{1}{\sqrt{2}}+O(e^{-c\epsilon
N})\right)\|h\|_{\infty,\Lambda}.$

Clearly, for $N$ large enough, we obtain the desired contradiction. Note that
this argument only works when $\dim(\rho)\geq 2.$ We also note that this is
only a very rough sketch of the proof; several additional arguments are
required to make it precise. Finally, the case where $\dim(\rho)=1$ requires
special treatment and is based on a compactness argument which is contained in
[21].

###  1.5. Acknowledgments

I would like to thank Michael Magee who pointed me to his joint work with Jean
Bourgain and Alex Kontorovich [20].

##  2\. Preliminaries

###  2.1. Schottky groups

Let us briefly review some properties of Schottky groups, referring the reader
to Borthwick’s book [3, Chapter 15] for a comprehensive discussion. The group
$\mathrm{SL}_{2}(\mathbb{R})$ acts on the extended complex plane
$\overline{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ by Möbius transformations

$\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R}),\,z\in\overline{\mathbb{C}}\Longrightarrow\gamma(z)=\frac{az+b}{cz+d}.$

A Schottky group is a convex cocompact subgroup
$\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$ constructed in the following way:

  * •

Fix $m\geq 1$ and non-intersecting euclidean disks
$D_{1},\dots,D_{2m}\subset\mathbb{C}$ with centers on the real line,

  * •

Fix a positive integer $m$ and non-intersecting euclidean disks
$D_{1},\dots,D_{2m}\subset\mathbb{C}$ (in no particular order) with centers on
the real line.

  * •

For every $j\in\\{1,\dots,m\\}$ let $\gamma_{j}\in\mathrm{SL}_{2}(\mathbb{R})$
be the isometry that maps the exterior of $D_{j}$ to the interior of $D_{j+m}$
and for every $j\in\\{m+1,\dots,2m\\}$ put

$\gamma_{j}\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{j-m}^{-1}.$

For notational purposes it is helpful to the define the indices cyclically
modulo $2m$ so that

$\gamma_{j+m}=\gamma_{j}^{-1}$ and $\gamma_{j+2m}=\gamma_{j}.$

  * •

Let $\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$ be the free group generated by
the elements $\gamma_{1},\dots,\gamma_{2m}.$

One of the standard models for the hyperbolic plane is the Poincaré half-plane

$\mathbb{H}^{2}=\\{x+iy\in\mathbb{C}\ :\ y>0\\}$

endowed with its standard metric of constant curvature $-1$,

$ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}.$

By Button’s result [13] every convex cocompact hyperbolic surface $X$ can be
realized as the quotient of $\mathbb{H}^{2}$ by a Schottky group $\Gamma$, see
also [3, Theorem 15.3]. Notice that the complement
$\mathbb{H}^{2}\smallsetminus\bigcup_{j=1}^{2m}D_{j}$ provides a fundamental
domain for the action of $\Gamma$ on $\mathbb{H}^{2}$. Given a convex
cocompact hyperbolic surface $X$ we fix a Schottky group $\Gamma$ such that
$X\cong\Gamma\backslash\mathbb{H}^{2}$ and we refer to $D_{1},\dots,D_{2m}$
and $\gamma_{1},\dots,\gamma_{2m}$ as the Schottky data of $X$ (or $\Gamma$)
and we refer to $\gamma_{1},\dots,\gamma_{2m}$ as the Schottky generators of
$\Gamma$.

$\partial\mathbb{H}^{2}$$\mathbb{H}^{2}$$D_{1}$$D_{4}$$D_{2}$$D_{3}$$D_{5}$$D_{6}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$
Figure 1. A configuration of Schottky disks and isometries with $m=3$

###  2.2. Selberg zeta function and transfer operator

Let $\Gamma<\mathrm{SL}_{2}(\mathbb{R})$ be a finitely generated Fuchsian
group. It is well-known that the set of prime periodic geodesics on
$X=\Gamma\backslash\mathbb{H}^{2}$ is bijective to the set
$[\Gamma]_{\mathsf{prim}}$ of $\Gamma$-conjugacy classes of the primitive
hyperbolic elements in $\Gamma$ (see for instance [3]). We denote by
$\ell(\gamma)$ the length of the geodesic corresponding to the conjugacy class
$[\gamma]\in[\Gamma]_{\mathsf{prim}}$.

Let $V$ be finite-dimensional complex vector space endowed with a hermitian
inner product $\langle\cdot,\cdot\rangle_{V}$ and induced norm
$\|\textbf{v}\|_{V}=\sqrt{\langle\textbf{v},\textbf{v}\rangle_{V}}$ and let
$\rho\colon\Gamma\to\mathrm{U}(V)$ be a unitary representation of $\Gamma$.
Then the twisted Selberg function is defined for $\mathrm{Re}(s)\gg 1$ by the
infinite product

(12)
$Z_{\Gamma}(s,\rho)\stackrel{{\scriptstyle\text{def}}}{{=}}\prod_{k=0}^{\infty}\prod_{[\gamma]\in[\Gamma]_{\mathsf{prim}}}\det\left(I_{V}-\rho(\gamma)e^{-(s+k)\ell(\gamma)}\right).$

Notice that (12) reduces to the classical Selberg zeta function when $\rho$ is
the trivial one-dimensional representation of $\Gamma.$

Now let $\Gamma$ be a Schottky group with Schottky data $D_{1},\dots,D_{2m}$
and $\gamma_{1},\dots,\gamma_{2m}$, which we assume to be fixed. Let

$D\stackrel{{\scriptstyle\text{def}}}{{=}}\bigsqcup_{j=1}^{2m}D_{j}$

be the union of the Schottky disks and consider the Hilbert space $H^{2}(D,V)$
of square-integrable, holomorphic functions on $D,$

(13) $H^{2}(D,V)\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{\text{$F\colon
D\to V$ holomorphic}\left|\;\left\|F\right\|<\infty\right.\right\\}.$

Here $\|F\|$ is the $L^{2}$-norm given by

$\|F\|^{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\int_{D}\|F(z)\|_{V}^{2}\operatorname{dvol}(z),$

where $\operatorname{vol}$ denotes the Lebesgue measure on the complex plane.
The choice of the functional space is not very important in the present paper;
we choose $H^{2}(D,V)$ mostly as a matter of definiteness. Finally, we define
the twisted transfer operator

(14) $\mathcal{L}_{s,\rho}\colon H^{2}(D,V)\to H^{2}(D,V)$

by the formula

(15)
$\mathcal{L}_{s,\rho}F(z)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{2m}\gamma_{i}^{\prime}(z)^{s}\rho(\gamma_{i})^{-1}F(\gamma_{i}(z))\
\mathrm{if}\ z\in D_{j}$

for all $F\in H^{2}(D,V)$. In the one-dimensional case $V=\mathbb{C}$, the
functional space (13) reduces to the classical Bergman space $H^{2}(D)$ and
the operator (14) reduces to the well-known transfer operator which can be
found for instance in Chapter 14 of [3]). The twisted Selberg zeta function is
related to the transfer operator $\mathcal{L}_{s,\rho}$ through the following
result:

######  Proposition 2.1 (Fredholm determinant identity).

For every $s\in\mathbb{C}$ the operator (14) is trace class and we have the
identity

(16) $Z_{\Gamma}(s,\rho)=\det(1-\mathcal{L}_{s,\rho}).$

Identities such as (16) are well-known in thermodynamic formalism, a subject
going back to Ruelle [38], at least when $\rho$ is the trivial, one-
dimensional representation. The relation between the Selberg zeta function and
transfer operators has been studied by a number of different authors. For the
convex cocompact setting (no cusps) we refer to [36, 37, 16]. The extension to
non-trivial twists $\rho$ can be found in the more recent papers [14, 30, 31,
29]. A proof of Proposition 2.1 can be found in [21].

Proposition 2.1 has the following remarkable corollary: since
$\mathcal{L}_{s,\rho}$ depends holomorphically on the variable $s$, it follows
directly that $Z_{\Gamma}(s,\rho)$ is an entire function when $\Gamma$ is a
Schottky group, which is far from obvious from its definition in (12) as an
infinite product over conjugacy classes.

###  2.3. Finite covers and Venkov–Zograf formula

Twisted Selberg zeta functions are extremely helpful when studying resonances
in families of covers of surfaces $X=\Gamma\backslash\mathbb{H}^{2}.$ Fix a
finite degree, regular cover
$X^{\prime}=\Gamma^{\prime}\backslash\mathbb{H}^{2}$ of $X.$ We may assume
without loss of generality that $\Gamma^{\prime}$ is a normal subgroup of
$\Gamma,$ in which case the quotient $G=\Gamma/\Gamma^{\prime}$ is a group,
called the covering group (sometimes also called the Galois group of the
covering). The (left) regular representation of $G$ is the representation
$\mathsf{R}_{G}\colon G\to\mathrm{U}(L^{2}(G))$ defined by

$\mathsf{R}_{G}(g)\varphi(x)=\varphi(xg^{-1}).$

for all $g\in G$, $x\in G$ and $\varphi\in L^{2}(G)$. This representation can
be extended in a natural way to a representation
$\mathsf{R}_{G}\colon\Gamma\to\mathrm{U}(L^{2}(G))$ by setting

(17)
$\mathsf{R}_{G}(\gamma)\stackrel{{\scriptstyle\text{def}}}{{=}}\mathsf{R}_{G}(\pi(\gamma))$

for all $\gamma\in\Gamma$, where $\pi\colon\Gamma\to G$ is the natural
projection map. To keep the notation simple, we will not distinguish the
regular representation $\mathsf{R}_{G}$ of $G$ from its extension to $\Gamma$.
We then have the relation

(18) $Z_{\Gamma^{\prime}}(s)=Z_{\Gamma}(s,\mathsf{R}_{G}),$

which was proven by Venkov–Zograf [42] in the case where $\Gamma$ is a
cofinite Fuchsian group (see also [41]). For an extension of this formula to
the non-cofinite case we refer to [14].

Twisted Selberg functions enjoy the following nice property: for any two
finite-dimensional unitary representations $(\rho_{1},V_{1})$ and
$(\rho_{2},V_{2})$ of $\Gamma$, we have the factorization

(19)
$Z_{\Gamma}(s,\rho_{1}\oplus\rho_{2})=Z_{\Gamma}(s,\rho_{1})Z_{\Gamma}(s,\rho_{2}),$

where the symbol $\oplus$ stands for the direct sum of representations. As is
well-known, the regular representation decomposes as the direct sum

(20) $\mathsf{R}_{G}=\bigoplus_{(\rho,V)\in\mathsf{irrep}(G)}\dim(\rho)\rho,$

where the direct sum is over the set $\mathsf{irrep}(G)$ of all irreducible
representations $(\rho,V)$ of $\Gamma$ (up to equivalence of representations).
Keeping in mind that each representation $\rho$ of $G$ extends to a
representation of $\Gamma$ through

(21)
$\rho(\gamma)\stackrel{{\scriptstyle\text{def}}}{{=}}\rho(\pi(\gamma)),\quad\gamma\in\Gamma,$

we obtain the identity

(22)
$Z_{\Gamma^{\prime}}(s)=\prod_{\rho\in\mathsf{irrep}(G)}Z_{\Gamma}(s,\rho)^{\dim(\rho)},$

or equivalently,

(23)
$Z_{\Gamma^{\prime}}(s)=Z_{\Gamma}(s)\prod_{\rho\in\mathsf{irrep}_{0}(G)}Z_{\Gamma}(s,\rho)^{\dim(\rho)},$

where the product on the right of (23) is over all non-trivial, irreducible
representations of $G.$ We will refer to (22) and (23) as the Venkov–Zograf
(factorization) formula.

###  2.4. Reduced words and bounds for derivatives

From now on, we assume that $X=\Gamma\backslash\mathbb{H}^{2}$ is a fixed
convex cocompact hyperbolic surface and that $\Gamma$ is a Schottky group with
Schottky data $D_{1},\dots,D_{2m}$ and $\gamma_{1},\dots,\gamma_{2m}$. We will
henceforth drop $\Gamma$ from the subscripts, writing $\delta=\delta_{\Gamma}$
and $\Lambda=\Lambda_{\Gamma}.$ Moreover, we let $\mathcal{L}_{s,\rho}$ be the
transfer operator introduced in (15).

We will now introduce additional notations which will be used throughout this
work. Given a finite word

$\textbf{a}=a_{1}\cdots a_{N}$

with
$a_{1},\dots,a_{N}\in[2m]\stackrel{{\scriptstyle\text{def}}}{{=}}\\{1,\dots,2m\\}$,
we set

$\gamma_{\textbf{a}}\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{a_{1}}\circ\cdots\circ\gamma_{a_{N}}.$

Recall that $\Gamma$ is freely generated by the elements
$\gamma_{1},\dots,\gamma_{2m}$, that is, there are no relations among the
elements $\gamma_{1},\dots,\gamma_{2m}$ other than the trivial relations
$\gamma_{i}^{-1}\gamma_{i}=\gamma_{i}\gamma_{i}^{-1}=e$. Recall also that
$\gamma_{i}^{-1}=\gamma_{i+m},$ where the indices are defined modulo $2m$. We
say that the word a is reduced if the corresponding element
$\gamma_{\textbf{a}}$ is reduced when viewed as a word in the alphabet
$\\{\gamma_{1},\dots,\gamma_{2m}\\}$. We then denote by $W_{N}$ the set of
reduced words of length $N$,

(24) $W_{N}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\textbf{a}=a_{1}\cdots
a_{N}:\text{ $a_{i}\neq a_{i+1}+m\;(\mathrm{mod}\;2m)$ for all
$i=1,\dots,N-1$}\\},$

which corresponds one-to-one to the elements of reduced word length $N$ in
$\Gamma$ via the map $\textbf{a}\mapsto\gamma_{\textbf{a}}.$ Moreover, we
define for all $j\in[2m]$ the set

$W_{N}^{j}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\textbf{a}=a_{1}\cdots
a_{N}\in W_{N}:a_{N}\neq j\\}.$

Observe that if $\textbf{a}=a_{1}\cdots a_{N}\in W_{N}^{j},$ then
$\gamma_{\textbf{a}}$ maps the closure of $D_{j}$ into the interior
$D_{a_{1}+m}$. With these notations in place, we have for all $N\in\mathbb{N}$
and all $j\in[2m]$ the formula

(25) $\mathcal{L}_{s,\rho}^{N}F(z)=\sum_{\textbf{a}\in
W_{N}^{j}}\gamma_{\textbf{a}}^{\prime}(z)^{s}\rho(\gamma_{\textbf{a}})^{-1}F(\gamma_{\textbf{a}}(z))\text{
if $z\in D_{j}$}.$

Finally, we record two distortion estimates (see [28] for the proofs) which
are crucial in this work:

  * •

(Uniform hyperbolicity) There exist constants $c_{1},c_{2}>0$ and
$0<\theta_{1}<\theta_{2}<1$ such that for all $N$, all $j\in[2m]$ and all
$\textbf{a}\in W_{N}^{j}$, we have

(26) $c_{1}\theta_{1}^{N}<\sup_{z\in
D_{j}}|\gamma^{\prime}_{\textbf{a}}(z)|<c_{2}\theta_{2}^{N}.$

  * •

(Bounded distortion) There exists a constant $c_{3}>0$ such that for all $N$,
all $j\in[2m]$, all $\textbf{a}\in W_{N}^{j}$ and all $z\in D_{j}$ we have

(27) $\sup_{z\in
D_{j}}\left|\frac{\gamma_{\textbf{a}}^{\prime\prime}(z)}{\gamma_{\textbf{a}}^{\prime}(z)}\right|\leq
c_{3}.$

The bounded distortion estimate has the following important consequence: there
exists a constant $c_{5}>0$ such that for all words $\textbf{a}\in W_{N}^{j}$
and all pair of points $z_{1},z_{2}\in D_{j}$, we have

(28)
$\left|\frac{\gamma_{\textbf{a}}^{\prime}(z_{1})}{\gamma_{\textbf{a}}^{\prime}(z_{2})}\right|\leq
c_{4}.$

###  2.5. Topological pressure and Ruelle-Perron-Frobenius theorem

Let $D_{1},\dots,D_{2m}$ and $\gamma_{1},\dots,\gamma_{2m}$ be the Schottky
data of $\Gamma.$ For all $j\in[2m]$ let
$I_{j}\stackrel{{\scriptstyle\text{def}}}{{=}}D_{j}\cap\mathbb{R}$ be the real
interval spanned by the Schottky disk $D_{j}.$ It turns out that the map

$T\colon
I\stackrel{{\scriptstyle\text{def}}}{{=}}\bigsqcup_{j=1}^{2m}I_{j}\rightarrow
I,\quad T(x)=\gamma_{j}(x)\ \mathrm{if}\ x\in I_{j},$

sometimes called the Bowen–Series map, encodes the dynamics of the full group
$\Gamma$. The origins of this type of coding go back to the work of
Bowen–Series [9]. The limit set $\Lambda$ can be re-interpreted as the non-
wandering set of the map $T$:

(29) $\Lambda=\bigcap_{N=1}^{\infty}T^{-N}(I).$

Given any continuous function $\varphi\colon I\to\mathbb{R}$, we define the
topological pressure in terms of weighted sums over periodic orbits through
the formula

(30)
$P(\varphi)\stackrel{{\scriptstyle\text{def}}}{{=}}\lim_{N\rightarrow\infty}\frac{1}{N}\log\left(\sum_{T^{N}x=x}e^{\varphi^{(n)}(x)}\right),$

where

(31)
$\varphi^{(n)}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\varphi(x)+\varphi(Tx)+\ldots+\varphi(T^{n-1}x).$

By the variational formula we have

(32) $P(\varphi)=\sup_{\mu}\left(h_{\mu}(T)-\int_{\Lambda}\varphi
d\mu\right),$

where $\mu$ ranges over the set of $T$-invariant probability measures and
$h_{\mu}(T)$ is the measure theoretic entropy. We refer the reader to [33] for
general facts on topological pressure and thermodynamical formalism. More
important for us is Bowen’s celebrated result [8] which says that the map

(33) $\mathbb{R}\to\mathbb{R},\quad\sigma\mapsto
P(\sigma)\stackrel{{\scriptstyle\text{def}}}{{=}}P(-\sigma\log|T^{\prime}|)$

is convex, strictly decreasing and vanishes precisely at $\sigma=\delta$, the
Hausdorff dimension of the limit set. The relevance of the topological
pressure stems from the Ruelle-Perron-Frobenius theorem which we recall here:

######  Proposition 2.2 (Ruelle-Perron-Frobenius).

Set $\mathcal{L}_{\sigma}=\mathcal{L}_{\sigma,\mathrm{id}}$ where
$\sigma\in\mathbb{R}$ is real and $\operatorname{id}$ is the one-dimensional
trivial representation. Then the following statements hold true:

  * •

The spectral radius of $\mathcal{L}_{\sigma}$ on $C^{1}(I)$ is equal to
$e^{P(\sigma)}$ which is a simple eigenvalue associated to a strictly positive
eigenfunction $\varphi_{\sigma}>0$ in $C^{1}(I)$.

  * •

The operator $\mathcal{L}_{\sigma}$ on $C^{1}(I)$ is quasi-compact with
essential spectral radius smaller than $\kappa(\sigma)e^{P(\sigma)}$ for some
$\kappa(\sigma)<1$.

  * •

There are no other eigenvalues on $|z|=e^{P(\sigma)}$.

  * •

Moreover, the spectral projector $\mathbb{P}_{\sigma}$ on
$\\{e^{P(\sigma)}\\}$ is given by

$\mathbb{P}_{\sigma}(f)=\varphi_{\sigma}\int_{\Lambda}fd\mu_{\sigma},$

where $\mu_{\sigma}$ is the unique $T$-invariant probability measure on
$\Lambda$ that satisfies

$\mathcal{L}_{\sigma}^{*}(\mu_{\sigma})=e^{P(\sigma)}\mu_{\sigma}.$

For a proof, we refer to [33, Theorem 2.2]. An important consequence of the
Ruelle–Perron–Frobenius theorem is the following estimate due to Naud [28]:

######  Lemma 2.3 (Pressure estimate).

For every $M>0$ there exists a constant $C>0$ such that for all
$\sigma\in[0,M]$ and all $N\geq 1$ we have

(34) $\sum_{j=1}^{2m}\sum_{\textbf{a}\in
W_{N}^{j}}\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}^{\sigma}\leq
Ce^{NP(\sigma)},$

where we use the notation

$\|g\|_{\infty,D_{j}}\stackrel{{\scriptstyle\text{def}}}{{=}}\sup_{z\in
D_{j}}|g(z)|,$

and $P(\sigma)$ is the pressure function defined in (33).

##  3\. Exploiting the expansion property

###  3.1. Exponential decay estimate for $W_{N}$

We will continue with the notations as before. Let $\Gamma$ be a non-
elementary Schottky group with generators $\gamma_{1},\dots,\gamma_{m},$ let
$G$ be a finite group and let $\pi\colon\Gamma\to G$ be a homorphism. It is
important to notice for every representation $(\nu,W)$ of $G$, we can endow
its representation space $W$ with a hermitian inner product
$\langle\cdot,\cdot\rangle_{W}$ with respect to which $\nu(g)$ is unitary for
every $g\in G.$ We will always tacitly assume that $W$ is endowed with such an
inner product and we let

$\|\textbf{w}\|_{W}\stackrel{{\scriptstyle\text{def}}}{{=}}\sqrt{\langle\textbf{w},\textbf{w}\rangle_{W}}$

be the induced norm. Recall that every representation $(\nu,W)$ of $G$ extends
naturally to a representation $(\nu,W)$ of $\Gamma$ by

$\nu(\gamma)\stackrel{{\scriptstyle\text{def}}}{{=}}\nu(\pi(\gamma)),\quad\gamma\in\Gamma.$

Furthermore, for every finite set $Z$ of words in the alphabet
$[2m]=\\{1,\dots,2m\\}$, we define the operator

$Z(\nu)\colon W\to W,\quad
Z(\nu)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{\textbf{a}\in
Z}\nu(\gamma_{\textbf{a}})$

Finally, we define the operator norm of a linear operator $\textbf{A}\colon
W\to W$ by

(35)
$\|\textbf{A}\|_{\mathrm{End}(W)}\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{\textbf{w}\in
W\smallsetminus\\{0\\}}\frac{\|\textbf{A}\textbf{w}\|_{W}}{\|\textbf{w}\|_{W}}.$

With this in place we can now state the main result of this section.

######  Proposition 3.1 (Norm estimate for $W_{N}(\nu)$).

Assume that $\\{\pi(\gamma_{1}),\dots,\pi(\gamma_{m})\\}$ is an
$\epsilon$-expanding subset of $G$. Then for every representation $(\nu,W)$ of
$G$ not containing the trivial representation, we have

$\|W_{N}(\nu)\|_{\mathrm{End}(W)}\ll e^{-\frac{\epsilon}{4}N}|W_{N}|,$

where the implied constant depends on $m$ and $\epsilon$ but not on $N.$

###### Proof.

By decomposing $(\nu,W)$ into irreducibles, we may assume without loss of
generality that $(\nu,W)$ is a non-trivial, irreducible representation. Let
$T\colon L^{2}(G)\to L^{2}(G)$ be the operator given by

$Tf(x)=\frac{1}{2m}\sum_{j=1}^{m}\left(f(\pi(\gamma_{j})g)+f(\pi(\gamma_{j})^{-1}g)\right)$

for $f\in L^{2}(G)$ and $x\in G.$ Recall that since
$\\{\pi(\gamma_{1}),\dots,\pi(\gamma_{m})\\}$ is $\epsilon$-expanding, the
eigenvalues of $T,$ restricted to the subspace $L_{0}^{2}(G)$ of functions $f$
satisfying

$\sum_{x\in G}f(x)=0,$

are contained in $[-1+\epsilon,1-\epsilon].$ By decomposing the regular
representation of $G$ into irreducibles, we deduce that the eigenvalues
$\lambda_{1},\dots,\lambda_{d}$ of

$W_{1}(\nu)=\frac{1}{2m}\sum_{j=1}^{2m}\nu(\gamma_{j})=\frac{1}{2m}\sum_{j=1}^{m}\left(\nu(\gamma_{j})+\nu(\gamma_{j}^{-1})\right)\in\mathrm{End}(W)$

are contained in the spectrum of the operator $T$ restricted to
$L_{0}^{2}(G).$ Consequently, the absolute values of the eigenvalues of
$W_{1}(\nu)$ are bounded by

(36) $\max_{1\leq k\leq d}|\lambda_{k}|\leq 2m(1-\epsilon),$

which we will use later on in the proof. Now consider the formal power series
with coefficients in the endomorphism ring $\mathrm{End}(W),$

(37)
$G_{\nu}(z)\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{N=0}^{\infty}W_{N}(\nu)z^{N},$

where $W_{0}(\nu)\stackrel{{\scriptstyle\text{def}}}{{=}}I_{W}$ is the
identity operator on $W.$ An elementary counting argument shows that

(38) $|W_{N}|\asymp(2m-1)^{N},$

where the implied constants are independent of $N.$ Thus we can trivially
estimate the norm of $W_{N}(\nu)$ as

$\|W_{N}(\nu)\|_{\mathrm{End}(W)}\leq|W_{N}|\ll(2m-1)^{N}.$

We deduce that (37) is absolutely convergent for $|z|<\frac{1}{2m-1}$ with
respect to the operator norm $\|\cdot\|_{\mathrm{End}(W)}.$ Next, we derive a
closed-form expression for $G_{\nu}(z)$. To that effect, note that we have the
elementary but essential recursion formulas222These recursion formulas were
used by Ihara [19] in his derivation of a closed-form expression of what came
to be known as the Ihara zeta function:

(39) $W_{1}(\nu)^{2}=W_{2}(\nu)+2m\cdot I_{W}$

and for all $N\geq 2$,

(40) $W_{1}(\nu)W_{N}(\nu)=W_{N+1}(\nu)+(2m-1)W_{N-1}(\nu).$

We can now multiply $G_{\nu}(z)$ by $zW_{1}(\nu)$ on the right and apply the
recursion formulas (39) and (40) to obtain

$\displaystyle zW_{1}(\nu)G_{\nu}(z)$
$\displaystyle=W_{1}(\nu)z+W_{1}(\nu)^{2}z^{2}+\sum_{N=2}^{\infty}W_{1}(\nu)W_{N}(\nu)z^{N+1}$
$\displaystyle=W_{1}(\nu)z+(W_{2}(\nu)+2m\cdot I_{W})z^{2}$
$\displaystyle\qquad\qquad+\sum_{N=2}^{\infty}W_{N+1}(\nu)z^{N+1}+(2m-1)\sum_{N=2}^{\infty}W_{N-1}(\nu)z^{N+1}.$

Inserting

$\sum_{N=2}^{\infty}W_{N+1}(\nu)z^{N+1}=G_{\nu}(z)-W_{2}(\nu)z^{2}-W_{1}(\nu)z-I_{W}$

and

$\sum_{N=2}^{\infty}W_{N-1}(\nu)z^{N+1}=z^{2}\left(G_{\nu}(z)-I_{W}\right)$

into the above equation yields

(41)
$zW_{1}(\nu)G_{\nu}(z)=(z^{2}-1)I_{W}+G_{\nu}(z)\left(1+(2m-1)z^{2}\right).$

Rearranging (41) gives the closed-form expression

$G_{\nu}(z)=(1-z^{2})\left(I_{W}-W_{1}(\nu)z+(2m-1)z^{2}\right)^{-1}.$

Now observe that $W_{1}(\nu)$ is a self-adjoint operator by virtue of the fact
that $\nu$ is a unitary representation and $W_{1}$ is symmetric (that is,
$W_{1}^{-1}=W_{1}$). Thus there exists a basis of $W$ with respect to which
$W_{1}(\nu)$ acts by the diagonal matrix

$W_{1}(\nu)=\operatorname{diag}(\lambda_{1},\dots,\lambda_{d}).$

For each $k\in[d]$ let $\omega^{\pm}_{k}$ be the pair of complex numbers
satisfying

$1-\lambda_{k}z+(2m-1)z^{2}=(1-\omega_{k}^{+}z)(1-\omega_{k}^{-}z).$

With respect to the diagonalizing basis, we then have the expression

(42)
$G_{\nu}(z)=\operatorname{diag}\left(\frac{1-z^{2}}{(1-\omega_{k}^{+}z)(1-\omega_{k}^{-}z)}\right)_{k=1}^{d}.$

Expanding each diagonal entry as a power series in $z$ gives

$\frac{1-z^{2}}{(1-\omega_{k}^{+}z)(1-\omega_{k}^{-}z)}=\sum_{N=0}^{\infty}(\xi_{k,N}-\xi_{k,N-2})z^{N},$

where $\xi_{k,-1}=\xi_{k,-2}=0$ and

(43)
$\xi_{k,N}=\sum_{l=0}^{N}\left(\omega_{k}^{+}\right)^{l}\left(\omega_{k}^{-}\right)^{N-l}$

for all non-negative integers $N$. By comparing coefficients in (42) and (37),
we find

$W_{N}(\nu)=\operatorname{diag}\left(\xi_{k,N}-\xi_{k,N-2}\right)_{k=1}^{d}.$

It follows that

(44) $\|W_{N}(\nu)\|_{\mathrm{End}(W)}=\max_{1\leq k\leq
d}|\xi_{k,N}-\xi_{k,N-2}|.$

A direct computation shows that

(45)
$\omega_{k}^{\pm}=\frac{\lambda_{k}\pm\sqrt{\lambda_{k}^{2}-4(2m-1)}}{2}.$

If $|\lambda_{k}|\leq 2\sqrt{2m-1}$ it follows directly from (45) that

$|\omega_{k}^{\pm}|=\sqrt{2m-1}.$

Now suppose $|\lambda_{j}|>2\sqrt{2m-1}$. In that case we recall from (36)
that

$|\lambda_{j}|\leq 2m(1-\epsilon).$

We deduce

$|\omega_{k}^{\pm}|\leq\frac{|\lambda_{k}|+\sqrt{(2m)^{2}-4(2m-1)}}{2}\leq(2m-1)\left(1-\frac{m}{2m-1}\epsilon\right).$

Therefore, setting

$\epsilon^{\prime}=\min\left\\{\frac{m}{2m-1}\epsilon,\,1-\frac{1}{\sqrt{2m-1}}\right\\},$

we obtain

(46) $\max_{1\leq k\leq
d}|\omega_{k}^{\pm}|\leq(2m-1)(1-\epsilon^{\prime})<(2m-1)e^{-\epsilon^{\prime}}.$

Note that $0<\epsilon\leq 1,$ so we have

$\epsilon^{\prime}\geq\frac{\epsilon}{3}$

and consequently

(47) $\max_{1\leq k\leq
d}|\omega_{k}^{\pm}|\leq(2m-1)e^{-\frac{\epsilon}{3}}.$

Using (43), (47) and (38) we obtain

$\max_{1\leq k\leq d}|\xi_{k,N}|\leq N\left(\max_{1\leq k\leq
d}|\omega_{k}^{\pm}|\right)^{N}\leq Ne^{-\frac{\epsilon}{3}N}(2m-1)^{N}\ll
e^{-\frac{\epsilon}{4}N}|W_{N}|.$

Thus, returning to (44), we obtain

$\|W_{N}(\nu)\|_{\mathrm{End}(W)}\leq\max_{1\leq k\leq
d}|\xi_{k,N}|+\max_{1\leq k\leq d}|\xi_{k,N-2}|\ll
e^{-\frac{\epsilon}{4}N}|W_{N}|.$

This completes the proof of Proposition 3.1. ∎

For every positive integer $N$ and for every pair of indices $i,j\in[2m]$
consider the following subset of reduced words of length $N$:

(48) $A_{N}^{i,j}=\\{\textbf{a}=a_{1}\cdots a_{N}\in
W_{N}:a_{1}=i,a_{N}=j\\}.$

It turns out that in our application it is more useful to have a bound on the
operator norm of $A_{N}^{i,j}(\nu)$ rather than $W_{N}(\nu)$.

######  Corollary 3.2 (Norm estimate for $A_{N}^{i,j}(\nu)$).

Assume that $\\{\pi(\gamma_{1}),\dots,\pi(\gamma_{m})\\}$ is an
$\epsilon$-expanding subset of $G$. Then for every representation $(\nu,W)$ of
$G$ not containing the trivial representation, for every $N$ and for every
pair of indices $i,j\in[2m]$, we have the bound

$\|A_{N}^{i,j}(\nu)\|_{\mathrm{End}(W)}\ll e^{-\frac{\epsilon}{6}N}|W_{N}|$

where the implied constant depend only on $m$ and $\epsilon$.

###### Proof.

Consider first the set of reduced words of length $N$ starting with the letter
$i,$

$B_{N}^{i}=\\{\textbf{a}=a_{1}\cdots a_{N}\in W_{N}:a_{1}=i\\}.$

Observe that

$B_{N}^{i}(\nu)=\nu(\gamma_{i})\left(W_{N-1}(\nu)-B_{N-1}^{i+m}(\nu)\right).$

Applying the triangle inequality and using the fact that $\nu$ is unitary
gives

(49)
$\|B_{N}^{i}(\nu)\|_{\mathrm{End}(W)}\leq\|W_{N-1}(\nu)\|_{\mathrm{End}(W)}+\|B_{N-1}^{i+m}(\nu)\|_{\mathrm{End}(W)}.$

Hence, setting

(50) $\mathfrak{b}_{N}(\nu)\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{1\leq
i\leq 2m}\|B_{N}^{i}(\nu)\|_{\mathrm{End}(W)},$

we obtain

$\mathfrak{b}_{N}(\nu)\leq\|W_{N-1}(\nu)\|_{\mathrm{End}(W)}+\mathfrak{b}_{N-1}(\nu).$

Iterating this inequality gives

(51)
$\mathfrak{b}_{N}(\nu)\leq\sum_{k=0}^{N-1}\|W_{k}(\nu)\|_{\mathrm{End}(W)}.$

By Proposition 3.1 we have

$\|W_{k}(\nu)\|_{\mathrm{End}(W)}\ll e^{-\frac{\epsilon}{4}k}|W_{k}|\ll
e^{-\frac{\epsilon}{4}k}(2m-1)^{k},$

where the implied constant depends only on $m$. By inserting this bound into
(51), and using a geometric series summation and the fact that
$|W_{N}|\asymp(2m-1)^{N}$, we obtain

(52)
$\mathfrak{b}_{N}(\nu)\ll\sum_{k=0}^{N-1}e^{-\frac{\epsilon}{4}k}(2m-1)^{k}\ll
e^{-\frac{\epsilon}{5}N}(2m-1)^{N}\ll e^{-\frac{\epsilon}{5}N}|W_{N}|.$

Notice that we also have the relation

(53)
$A_{N}^{i,j}(\nu)=\left(B_{N-1}^{i}(\nu)-A_{N}^{i,j+m}(\nu)\right)\nu(\gamma_{j}).$

Now we set

$\mathfrak{a}_{N}(\nu)\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{1\leq
i,j\leq 2m}\|A_{N}^{i,j}(\nu)\|_{\mathrm{End}(W)}.$

and we repeat the argument above (with the necessary modifications). From (53)
we get the estimate

$\mathfrak{a}_{N}(\nu)\leq\|B_{N-1}^{i}(\nu)\|_{\mathrm{End}(W)}+\mathfrak{a}_{N-1}(\nu)\leq\mathfrak{b}_{N-1}(\nu)+\mathfrak{a}_{N-1}(\nu),$

which we can iterate to obtain

$\mathfrak{a}_{N}(\nu)\leq\sum_{k=0}^{N-1}\mathfrak{b}_{k}(\nu).$

Combining this with the bound in (52), we obtain

$\mathfrak{a}_{N}(\nu)\leq\sum_{k=0}^{N-1}e^{-\frac{\epsilon}{5}k}|W_{k}|\ll
e^{-\frac{\epsilon}{6}N}|W_{N}|,$

concluding the proof of Corollary 3.2. ∎

###  3.2. Exponential decay for twisted averages

For every function $h\colon\Lambda\to W$ we let

$\|h\|_{\infty,\Lambda}\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{x\in\Lambda}\|h(x)\|_{W}$

be the maximum norm on $\Lambda$ and we introduce the semi-norm

(54) $[h]_{\Lambda}\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{1\leq j\leq
2m}\max_{\begin{subarray}{c}x,y\in\Lambda\cap I_{j}\\\ x\neq
y\end{subarray}}\frac{\left\|h(x)-h(y)\right\|_{W}}{|x-y|},$

where $I_{j}=D_{j}\cap\mathbb{R}$ are the real intervals spanned by the
Schottky disks $D_{1},\dots,D_{2m}$ of $\Gamma$. Recall also that we use
$\mathbb{E}_{a\in A}f(a)$ as a shorthand for the average

$\frac{1}{|A|}\sum_{a\in A}f(a).$

The aim of this subsection is to prove the following key result:

######  Proposition 3.3 (Exponential decay for twisted averages).

Assume that $\\{\pi(\gamma_{1}),\dots,\pi(\gamma_{m})\\}$ is
$\epsilon$-expanding, let $(\nu,W)$ be a representation of $G$, and let
$h\colon\Lambda\to W$ be a function satisfying

(55) $\sum_{g\in G}\nu(g)h(x)=0$

for all $x\in\Lambda.$ Then, for every $j\in[2m]$ and for every $x\in\Lambda$
such that $x\in\Lambda\cap I_{j},$ we have

$\left\|\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h(\gamma_{\textbf{a}}(x))\right)\right\|_{W}\ll
e^{-c\epsilon N}\left(\|h\|_{\infty,\Lambda}+[h]_{\Lambda}\right)$

where the implied constant and $c>0$ depend solely on $\Gamma.$

###### Proof.

Fix $j\in[2m]$ and $x\in\Lambda$ such that $x\in\Lambda\cap I_{j}$ and a
function $h\colon\Lambda\to W$ satisfying (55). By decomposing $W$ into a
direct sum of irreducibles, we may assume without loss of generality that
$\nu$ is an irreducible representation of $G.$ Note that the linear map

$W\to W,\quad\textbf{v}\mapsto\frac{1}{|G|}\sum_{g\in G}\nu(g)\textbf{v}$

is the projection onto the subspace

$W_{\mathrm{triv}}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\textbf{w}\in
W:\text{$\nu(g)\textbf{w}=\textbf{w}$ for all $g\in G$}\\}.$

Thus, thanks to the condition in (55), we may assume furthermore that $\nu$ is
a non-trivial, irreducible representation.

For each pair $j,l\in\\{1,\dots,2m\\}$ we define the set

$Z_{N}^{l,j}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{\textbf{a}=a_{1}\cdots
a_{N}\in W_{N}:\text{$a_{1}=l+m$ and $a_{N}\neq j$}\\},$

which corresponds precisely to the set of reduced words $\textbf{a}\in W_{N}$
for which $\gamma_{\textbf{a}}$ maps the disk $D_{j}$ to the interior of
$D_{l}$. Note that

$Z_{N}^{l,j}=\bigsqcup_{\begin{subarray}{c}1\leq j^{\prime}\leq 2m\\\
j^{\prime}\neq j\end{subarray}}A_{N}^{l+m,j^{\prime}},$

where $A_{N}^{i,j}$ are the sets that we have introduced in (48). Thus, by
Corollary 3.2, we have the estimate

$\left\|Z_{N}^{l,j}(\nu)\right\|_{\mathrm{End}(W)}\leq\sum_{\begin{subarray}{c}1\leq
j^{\prime}\leq 2m\\\ j^{\prime}\neq
j\end{subarray}}\left\|A_{N}^{l+m,j^{\prime}}(\nu)\right\|_{\mathrm{End}(W)}\ll
e^{-\frac{\epsilon}{6}N}|W_{N}|.$

Hence, since the sets $Z_{N}^{l,j}$ and $W_{N}^{j}$ are of comparable size, we
deduce the estimate

(56) $\left\|\mathbb{E}_{\textbf{a}\in
Z_{N}^{l,j}}\left(\nu(\gamma_{\textbf{a}})\right)\right\|_{\mathrm{End}(W)}=\frac{1}{|Z_{N}^{l,j}|}\left\|Z_{N}^{l,j}(\nu)\right\|_{\mathrm{End}(W)}\ll\frac{1}{|W_{N}^{j}|}\left\|Z_{N}^{l,j}(\nu)\right\|_{\mathrm{End}(W)}\ll
e^{-\frac{\epsilon}{6}N},$

which we will use later on.

We will use a decoupling argument similar to the one contained in [6, Section
8]. Write $N=N_{1}+N_{2},$ where $N_{1}$ and $N_{2}$ will be chosen further
below, and notice that for every $j\in[2m]$ and every word $\textbf{a}\in
W_{N}^{j}$ there exists a unique index $l\in[2m]$ and unique words
$\textbf{a}_{1}\in W_{N_{1}}^{l}$ and $\textbf{a}_{2}\in Z_{N_{2}}^{l,j}$ with
$\textbf{a}=\textbf{a}_{1}\textbf{a}_{2}$. (Here,
$\textbf{a}_{1}\textbf{a}_{2}$ stands for the concatenation of the words
$\textbf{a}_{2}$ and $\textbf{a}_{2}$.) Observe also that

$|W_{N}^{j}|\asymp|W_{N_{1}}^{l}||Z_{N_{2}}^{l,j}|$

with implied constants depending only on $m.$ With this in mind, we can
estimate

(57) $\left\|\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h(\gamma_{\textbf{a}}(x))\right)\right\|_{W}\ll\sum_{l=1}^{2m}S_{l},$

where

$S_{l}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{1}\textbf{a}_{2}})^{-1}h(\gamma_{\textbf{a}_{1}\textbf{a}_{2}}(x))\right)\right\|_{W}.$

Further, we decompose the average

$\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{1}\textbf{a}_{2}})^{-1}h(\gamma_{\textbf{a}_{1}\textbf{a}_{2}}(x))\right)$

as

$\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left[\nu(\gamma_{\textbf{a}_{1}\textbf{a}_{2}})^{-1}\left(h(\gamma_{\textbf{a}_{1}\textbf{a}_{2}}(x))-\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right)\right]+\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{1}\textbf{a}_{2}})^{-1}\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right),$

and we use the triangle inequality to estimate

$S_{l}\leq S_{l}^{(1)}+S_{l}^{(2)},$

where

$S_{l}^{(1)}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left[\nu(\gamma_{\textbf{a}_{1}\textbf{a}_{2}})^{-1}\left(h(\gamma_{\textbf{a}_{1}\textbf{a}_{2}}(x))-\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right)\right]\right\|_{W}$

and

$S_{l}^{(2)}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{1}\textbf{a}_{2}})^{-1}\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right)\right\|_{W}.$

Note that for all points $x,y\in D_{j}$ belonging to the same Schottky disk,
we have

$\|h(\gamma_{\textbf{c}}(x))-h(\gamma_{\textbf{c}}(y))\|_{W}\ll\|\gamma_{\textbf{c}}^{\prime}\|_{\infty,D_{j}}[h]_{\Lambda}.$

Thus, applying the triangle inequality again, we can further estimate

$\displaystyle S_{l}^{(1)}$ $\displaystyle\leq\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}\big{(}h(\gamma_{\textbf{a}_{1}\textbf{a}_{2}}(x))-h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\big{)}\right\|_{W}$
$\displaystyle\leq\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}\left\|h(\gamma_{\textbf{a}_{1}\textbf{a}_{2}}(x))-h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right\|_{W}$
$\displaystyle\ll\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}\|\gamma_{\textbf{a}_{1}}^{\prime}\|_{\infty}[h]_{\Lambda}$
$\displaystyle=\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\|\gamma_{\textbf{a}_{1}}^{\prime}\|_{\infty}[h]_{\Lambda}.$

Using the pressure estimate from Lemma 2.3, we finally obtain

(58) $S_{l}^{(1)}\ll\frac{e^{N_{1}P(1)}}{|W_{N_{1}}^{l}|}[h]_{\Lambda}\ll
e^{-c^{\prime}N_{1}}[h]_{\Lambda},$

where we have set
$c^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}-P(1)+P(0)>0.$

Let us now turn our attention to the average $S_{l}^{(2)}.$ Rewritting it and
using the unitarity of $\nu$, we get

$\displaystyle S_{l}^{(2)}$ $\displaystyle=\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{2}})^{-1}\nu(\gamma_{\textbf{a}_{1}})^{-1}\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right)\right\|_{W}$
$\displaystyle=\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{2}})^{-1}\right)\cdot\nu(\gamma_{\textbf{a}_{1}})^{-1}\cdot\left(\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right)\right\|_{W}$
$\displaystyle\leq\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{2}})^{-1}\right)\right\|_{\mathrm{End}(W)}\left\|\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right\|_{W}.$

Recall that the limit set $\Lambda$ is invariant under the action of $\Gamma.$
Thus, since $x$ belongs to $\Lambda,$ so does
$\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x).$ Hence we can apply the triangle
inequality to estimate

$\left\|\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right\|_{W}\leq\mathbb{E}_{\textbf{b}_{2}\in
Z_{N_{2}}^{l,j}}\left\|h(\gamma_{\textbf{a}_{1}\textbf{b}_{2}}(x))\right\|_{W}\leq\|h\|_{\infty,\Lambda},$

which when inserted above yields

$\displaystyle S_{l}^{(2)}$ $\displaystyle\leq\mathbb{E}_{\textbf{a}_{1}\in
W_{N_{1}}^{l}}\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{2}})^{-1}\right)\right\|_{\mathrm{End}(W)}\|h\|_{\infty,\Lambda}$
$\displaystyle=\left\|\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{2}})^{-1}\right)\right\|_{\mathrm{End}(W)}\|h\|_{\infty,\Lambda}.$

Moreover, since $\nu$ is unitary, we have

$\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\left(\nu(\gamma_{\textbf{a}_{2}})^{-1}\right)=\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\nu(\gamma_{\textbf{a}_{2}})^{\ast}=\left(\mathbb{E}_{\textbf{a}_{2}\in
Z_{N_{2}}^{l,j}}\nu(\gamma_{\textbf{a}_{2}})\right)^{\ast},$

where $\textbf{A}^{\ast}$ denotes the adjoint of
$\textbf{A}\in\mathrm{End}(W)$. Since the norm in (35) equals the norm of its
adjoint, we can invoke the bound in (56) to estimate

(59) $S_{l}^{(2)}\ll e^{-\frac{\epsilon}{6}N_{2}}\|h\|_{\infty,\Lambda}.$

Adding together (58) and (59) yields

(60) $S_{l}\leq S_{l}^{(1)}+S_{l}^{(2)}\ll
e^{-\frac{\epsilon}{6}N_{2}}\|h\|_{\infty,\Lambda}+e^{-c^{\prime}N_{1}}[h]_{\Lambda}.$

Finally, choosing $N_{1}=\lceil\frac{\epsilon}{6c^{\prime}+\epsilon}N\rceil$
and $N_{2}=N-N_{1},$ we obtain

(61) $S_{l}\ll e^{-\frac{c^{\prime}}{6c^{\prime}+\epsilon}\epsilon
N}\left(\|h\|_{\infty,\Lambda}+[h]_{\Lambda}\right)\ll e^{-c\epsilon
N}\left(\|h\|_{\infty,\Lambda}+[h]_{\Lambda}\right),$

for all $l\in[2m],$ where we have set

$c\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{c^{\prime}}{6c^{\prime}+1}\leq\frac{c^{\prime}}{6c^{\prime}+\epsilon}.$

Inserting the bound (61) into (57) completes the proof. ∎

##  4\. Proof of Main Theorem

###  4.1. Normalized transfer operator

We will henceforth fix a finite-dimensional, unitary representation $(\rho,V)$
of $\Gamma$ and we will write $s\in\mathbb{C}$ in Cartesian coordinates as
$s=\sigma+it,$ where $\sigma$ and $t$ are the real and imaginary parts of $s,$
respectively. Given $s=\sigma+it,$ we define the normalized transfer operator
$\widetilde{\mathcal{L}}_{s,\rho}$ by

(62)
$\widetilde{\mathcal{L}}_{s,\rho}\stackrel{{\scriptstyle\text{def}}}{{=}}e^{-P(\sigma)}\varphi_{\sigma}^{-1}\mathcal{L}_{s,\rho}\varphi_{\sigma},$

where $\varphi_{\sigma}$ is the eigenfunction associated to the maximal
eigenvalue of the operator $\mathcal{L}_{\sigma}$ furnished by the
Ruelle–Perron–Frobenius Theorem (Proposition 2.2). Note that since
$\varphi_{\sigma}$ is positive on the limit set $\Lambda,$ (62) defines an
operator

$\widetilde{\mathcal{L}}_{s,\rho}\colon C^{0}(\Lambda,V)\to C^{0}(\Lambda,V)$

on the set of $V$-valued functions on $\Lambda.$ For all positive integers
$N,$ for all indices $j\in[2m]$ and for all points $x\in\Lambda$ with
$x\in\Lambda\cap I_{j},$ we have

(63)
$\widetilde{\mathcal{L}}_{s,\rho}^{N}f(x)=e^{-NP(\sigma)}\varphi_{\sigma}^{-1}\mathcal{L}_{s,\rho}^{N}\varphi_{\sigma}f(x)=\sum_{\textbf{a}\in
W_{N}^{j}}w_{\textbf{a},\sigma}(x)\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x)),$

where the $w_{\textbf{a},\sigma}(x)$’s are positive weights given by

(64)
$w_{\textbf{a},\sigma}(x)=\frac{\varphi_{\sigma}(\gamma_{\textbf{a}}(x))\gamma_{\textbf{a}}^{\prime}(x)^{\sigma}}{\varphi_{\sigma}(x)e^{NP(\sigma)}}.$

We call the operator $\widetilde{\mathcal{L}}_{s,\rho}$ “normalized” because
it satisfies

$\widetilde{\mathcal{L}}_{\sigma,\operatorname{id}}\textbf{1}=\textbf{1},$

with 1 being the constant one function on $\Lambda.$ Equivalently, the weights
sum up to 1, that is, for every $j\in[2m]$ and for every $x\in\Lambda$ with
$x\in\Lambda\cap I_{j},$ we have

$\sum_{\textbf{a}\in W_{N}^{j}}w_{\textbf{a},\sigma}(x)=1.$

Thus, $\widetilde{\mathcal{L}}_{s,\rho}^{N}f(x)$ (note that this can be viewed
as a vector in $V$) lies in the convex hull of the vectors

(65)
$\textbf{v}_{\textbf{a}}\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\in
V,\quad\textbf{a}\in W_{N}^{j}.$

The following result is a consequence of the fact that the unit sphere in $V$
is strictly convex.

######  Proposition 4.1 (Strict convexity).

Let $(\rho,V)$ be a finite-dimensional, unitary representation of $\Gamma$ and
let $F\in H^{2}(D,V)$ be a 1-eigenfunction of the transfer operator
$\mathcal{L}_{s,\rho}$ with $\sigma\in[0,\delta]$. Set
$f=\varphi_{\sigma}^{-1}F$ and let $x\in\Lambda\cap I_{j}$ (with $j\in[2m]$)
be a point such that $\|f(x)\|_{V}=\|f\|_{\infty,\Lambda}.$ Then, for all
positive integers $N,$ the bound

$\left\|f(x)-\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right\|_{V}\leq
C^{N}\sqrt{\delta-\sigma}\|f\|_{\infty,\Lambda}$

holds true for all words $\textbf{a}\in W_{N}^{j},$ where $C$ is a positive
constant depending only on $\Gamma.$

###### Proof.

Since $F$ is a 1-eigenfunction of $\mathcal{L}_{s,\rho},$ we have

(66) $\mathcal{L}_{s,\rho}^{N}F=F,$

for all positive integers $N$. By normalizing (66), we obtain

(67) $\widetilde{\mathcal{L}}_{s,\rho}^{N}f=e^{-NP(\sigma)}f,$

where $f=\varphi_{\sigma}^{-1}F$ and $\widetilde{\mathcal{L}}_{s,\rho}$ is the
normalized operator in (62). We can rewrite this as follows: for every
$j\in[2m]$ and for every $x\in\Lambda$ such that $x\in\Lambda\cap I_{j}$ we
have

(68) $e^{-NP(\sigma)}f(x)=\sum_{\textbf{a}\in
W_{N}^{j}}w_{\textbf{a},\sigma}(x)\textbf{v}_{\textbf{a}},$

where the $w_{\textbf{a},\sigma}(x)$’s are the weights in (64) and the
$\textbf{v}_{\textbf{a}}$’s are as in (65).

In what follows, we assume that $x\in\Lambda\cap I_{j}$ is a point satisfying
$\|f(x)\|_{V}=\|f\|_{\infty,\Lambda}.$ Since the $w_{\textbf{a},\sigma}(x)$’s
are real, we can square the norm of both sides of (68) to obtain

(69)
$e^{-2NP(\sigma)}\|f\|_{\infty,\Lambda}^{2}=e^{-2NP(\sigma)}\|f(x)\|_{V}^{2}=\sum_{\textbf{a},\textbf{b}\in
W_{N}^{j}}w_{\textbf{a},\sigma}(x)w_{\textbf{b},\sigma}(x)\mathrm{Re}\langle\textbf{v}_{\textbf{a}},\textbf{v}_{\textbf{b}}\rangle_{V}.$

Using the relation

$\mathrm{Re}\langle\textbf{v}_{\textbf{a}},\textbf{v}_{\textbf{b}}\rangle_{V}=\frac{1}{2}(\|\textbf{v}_{\textbf{a}}\|_{V}^{2}+\|\textbf{v}_{\textbf{b}}\|_{V}^{2}-\|\textbf{v}_{\textbf{a}}-\textbf{v}_{\textbf{b}}\|_{V}^{2}).$

and the fact that $\|\textbf{v}_{\textbf{a}}\|_{V}\leq\|f\|_{\infty,\Lambda}$
for all $\textbf{a}\in W_{N}^{j}$ by the choice of $x,$ we get

(70)
$\mathrm{Re}\langle\textbf{v}_{\textbf{a}},\textbf{v}_{\textbf{b}}\rangle_{V}\leq\|f\|_{\infty,\Lambda}^{2}-\frac{1}{2}\|\textbf{v}_{\textbf{a}}-\textbf{v}_{\textbf{b}}\|_{V}^{2}.$

Inserting (70) into (69) and using the fact that the weights
$w_{\textbf{a},\sigma}(x)$ are positive reals summing up to $1,$ we obtain

$e^{-2NP(\sigma)}\|f\|_{\infty,\Lambda}^{2}\leq\|f\|_{\infty,\Lambda}^{2}-\frac{1}{2}\sum_{\textbf{a},\textbf{b}\in
W_{N}^{j}}w_{\textbf{a},\sigma}(x)w_{\textbf{b},\sigma}(x)\|\textbf{v}_{\textbf{a}}-\textbf{v}_{\textbf{b}}\|_{V}^{2},$

which we can rearrange to give

(71) $\sum_{\textbf{a},\textbf{b}\in
W_{N}^{j}}w_{\textbf{a},\sigma}(x)w_{\textbf{b},\sigma}(x)\|\textbf{v}_{\textbf{a}}-\textbf{v}_{\textbf{b}}\|_{V}^{2}\leq
2\left(1-e^{-2NP(\sigma)}\right)\|f\|_{\infty,\Lambda}^{2}.$

This implies that for all pairs of words $\textbf{a},\textbf{b}\in W_{N}^{j},$
we have the bound

(72)
$\|\textbf{v}_{\textbf{a}}-\textbf{v}_{\textbf{b}}\|_{V}\leq\frac{\sqrt{2\left(1-e^{-2NP(\sigma)}\right)}}{\min_{\textbf{c}\in
W_{N}^{j}}\left(w_{\textbf{c},\sigma}(x)\right)}\|f\|_{\infty,\Lambda}.$

Since $\varphi_{\sigma}$ is positive on $\Lambda$ and $\Lambda$ is compact, we
can bound $\varphi_{\sigma}$ from above and below by constants depending only
on $\Gamma.$ Moreover, we can choose these constants uniformly in
$\sigma\in[0,\delta].$ Hence, from the definition of the weights and the
uniform hyperbolicity property in (26), we deduce that

(73) $w_{\textbf{c},\sigma}(x)\geq C^{-N}$

for all words $\textbf{c}\in W_{N}^{j}$ for some constant $C=C(\Gamma)>0.$
Using the continuity of the topological pressure $\sigma\mapsto P(\sigma)$ and
$P(\delta)=0$ gives

(74) $1-e^{-2NP(\sigma)}=e^{-2NP(\delta)}-e^{-2NP(\sigma)}\leq
CN(\delta-\sigma)$

for all $\sigma\leq\delta.$ Inserting (73) and (74) into (72) yields
furthermore (after increasing the constant $C$ if necessary)

(75) $\|\textbf{v}_{\textbf{a}}-\textbf{v}_{\textbf{b}}\|_{V}\leq
C^{N}\sqrt{\delta-\sigma}\|f\|_{\infty,\Lambda}$

for all pairs of words $\textbf{a},\textbf{b}\in W_{N}^{j}.$ Now we write

$f(x)-\textbf{v}_{\textbf{a}}=\sum_{\textbf{b}\in
W_{N}^{j}}w_{\textbf{b},\sigma}(x)\left(\textbf{v}_{\textbf{b}}-\textbf{v}_{\textbf{a}}\right).$

Applying the triangle inequality in the line above and then using the bound in
(75) completes the proof. ∎

We continue with a basic a priori estimate.

######  Proposition 4.2 (Lasota-Yorke-type estimate).

For every $T_{0}>0$ there exist positive constants $c=c(\Gamma)$ and
$C=C(T_{0},\Gamma)$ such that for every function $f\colon\Lambda\to V$ and
every $s=\sigma+it$ with $\sigma\in[0,1]$ and $|t|<T_{0},$ we have

$[\widetilde{\mathcal{L}}_{s,\rho}^{N}f]_{\Lambda}\leq
C\left(\|f\|_{\infty,\Lambda}+e^{-(c+P(\sigma))N}[f]_{\Lambda}\right).$

###### Proof.

Fix $j\in[2m]$ and points $x,y\in\Lambda\cap I_{j}.$ Using the formula (63),
we write

$\widetilde{\mathcal{L}}_{s,\rho}^{N}f(x)-\widetilde{\mathcal{L}}_{s,\rho}^{N}f(y)=\sum_{\textbf{a}\in
W_{N}^{j}}\rho(\gamma_{\textbf{a}})^{-1}\left[w_{\textbf{a},\sigma}(x)\gamma_{\textbf{a}}^{\prime}(x)^{it}f(\gamma_{\textbf{a}}(x))-w_{\textbf{a},\sigma}(y)\gamma_{\textbf{a}}^{\prime}(y)^{it}f(\gamma_{\textbf{a}}(y))\right].$

We can rewrite the bracketed term on the right hand side as

(76)
$\left(w_{\textbf{a},\sigma}(x)\gamma_{\textbf{a}}^{\prime}(x)^{it}-w_{\textbf{a},\sigma}(y)\gamma_{\textbf{a}}^{\prime}(y)^{it}\right)f(\gamma_{\textbf{a}}(x))+w_{\textbf{a},\sigma}(y)\gamma_{\textbf{a}}^{\prime}(y)^{it}\left(f(\gamma_{\textbf{a}}(x))-f(\gamma_{\textbf{a}}(y))\right).$

We will estimate the terms above individually. Using the definition of the
weights $w_{\textbf{a},\sigma}(x)$ in (64), the uniform hyperbolicity in (26),
and the bounded distortion property, we obtain

(77)
$\left|w_{\textbf{a},\sigma}(x)\gamma_{\textbf{a}}^{\prime}(x)^{it}-w_{\textbf{a},\sigma}(y)\gamma_{\textbf{a}}^{\prime}(y)^{it}\right|\ll
e^{-NP(\sigma)}(1+|t|)\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}^{\sigma}|x-y|.$

By the mean value theorem, we obtain

(78)
$\|f(\gamma_{\textbf{a}}(x))-f(\gamma_{\textbf{a}}(y))\|_{V}\ll[f]_{\Lambda}\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}.$

Thus we have

(79) $\|\text{\eqref{difference_bracketed}}\|_{V}\ll
e^{-NP(\sigma)}\left((1+|t|)\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}^{\sigma}\|f\|_{\infty,\Lambda}+\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}^{\sigma+1}[f]_{\Lambda}\right)|x-y|,$

where the implied constant depends on $T_{0}$ and $\Gamma.$ Hence, using the
triangle inequality and the fact that $\rho$ is unitary with respect to the
norm $\|\cdot\|_{V},$ we obtain

$\displaystyle\left\|\widetilde{\mathcal{L}}_{s,\rho}^{N}f(x)-\widetilde{\mathcal{L}}_{s,\rho}^{N}f(y)\right\|_{V}$
$\displaystyle\leq\sum_{\textbf{a}\in
W_{N}^{j}}\left\|w_{\textbf{a},\sigma}(x)\gamma_{\textbf{a}}^{\prime}(x)^{it}f(\gamma_{\textbf{a}}(x))-w_{\textbf{a},\sigma}(y)\gamma_{\textbf{a}}^{\prime}(y)^{it}f(\gamma_{\textbf{a}}(y))\right\|_{V}$
$\displaystyle\ll\sum_{\textbf{a}\in
W_{N}^{j}}\left(e^{-NP(\sigma)}\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}^{\sigma}\|f\|_{\infty,\Lambda}+e^{-NP(\sigma)}\|\gamma_{\textbf{a}}^{\prime}\|_{\infty,D_{j}}^{\sigma+1}[f]_{\Lambda}\right)|x-y|.$

Using the pressure estimate from Lemma 2.3, we get furthermore

(80)
$\left\|\widetilde{\mathcal{L}}_{s,\rho}^{N}f(x)-\widetilde{\mathcal{L}}_{s,\rho}^{N}f(y)\right\|_{V}\ll\left(\|f\|_{\infty,\Lambda}+e^{NP(1+\sigma)-NP(\sigma)}[f]_{\Lambda}\right)|x-y|.$

Finally, since the topological pressure $\sigma\mapsto P(\sigma)$ is
decreasing and since $\sigma>0$ by assumption, we have $P(1+\sigma)\leq P(1),$
which when inserted into (80) yields the desired estimate with $c=-P(1)>0.$ ∎

We will actually only use the following consequence of Proposition 4.2, which
says essentially that the semi-norm of normalized 1-eigenfunctions is
controlled by the maximum norm.

######  Lemma 4.3.

For every $T_{0}>0$ there exists a constant $C=C(T_{0},\Gamma)>0$ such that
for every 1-eigenfunction $F$ of $\mathcal{L}_{s,\rho}$ with
$\sigma\in[0,\delta]$ and $|t|<T_{0},$ the function $f=\varphi_{\sigma}^{-1}F$
satisfies the bound

$[f]_{\Lambda}\leq C\|f\|_{\infty,\Lambda}.$

###### Proof.

Notations and hypotheses being as in the statement of the lemma, we have

$e^{-NP(\sigma)}f=\widetilde{\mathcal{L}}_{s,\rho}f.$

Hence, by Proposition 4.2, we have

$e^{-NP(\sigma)}[f]_{\Lambda}\leq
C^{\prime}\left(\|f\|_{\infty,\Lambda}+e^{-(c+P(\sigma))N}[f]_{\Lambda}\right)$

for some constant $C^{\prime}=C^{\prime}(T_{0},\Gamma),$ or equivalently,

$[f]_{\Lambda}\leq
C^{\prime}\left(e^{NP(\sigma)}\|f\|_{\infty,\Lambda}+e^{-cN}[f]_{\Lambda}\right).$

Choosing $N$ sufficiently large such that $C^{\prime}e^{-cN}<\frac{1}{2}$ and
rearranging gives

$[f]_{\Lambda}\leq 2C^{\prime}e^{NP(\sigma)}\|f\|_{\infty,\Lambda},$

completing the proof. ∎

###  4.2. Finishing the proof

We can now prove the main theorem. Fix a Schottky group $\Gamma$ with
generators $\gamma_{1},\dots,\gamma_{m},$ a finite group $G$, and a
homomorphism $\pi\colon G\to\Gamma$. Let $\Gamma^{\prime}=\ker(\pi)$ be the
kernel of $\pi$ and suppose that
$\\{\pi(\gamma_{1}),\dots,\pi(\gamma_{m})\\}\subseteq G$ is
$\epsilon$-expanding for some $\epsilon>0.$ We need to show that the Selberg
zeta function $Z_{\Gamma^{\prime}}(s)$ has no new zeros (with respect to
$\Gamma$) in the strip $\mathrm{Re}(s)\geq\delta-\eta$, where the strip size
$\eta>0$ depends only on $\epsilon$ and $\Gamma$.

By the result of Magee–Naud [29], it suffices to show that
$Z_{\Gamma^{\prime}}(s)$ has no new zeros inside the complex rectangle

(81)
$R_{\eta}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{s=\sigma+it\in\mathbb{C}:\text{$\delta-\eta\leq\sigma\leq\delta$
and $|t|\leq T_{0}$}\\},$

where $T_{0}>0$ is some constant depending only on $\Gamma.$

Furthermore, in view of the Venkov–Zograf formula (22), to prove the main
theorem it is enough to show that for every non-trivial, irreducible
representation $(\rho,V)$ of $G$, the associated twisted Selberg zeta function
$Z_{\Gamma}(s,\rho)$ has no zeros inside $R_{\eta}$. We will first consider
the case where the dimension of $\rho$ is at least 2 (that is, the case where
$\rho$ is not a character).

#### The case $\dim(\rho)\geq 2$

Suppose that there exists an irreducible representation $(\rho,V)$ of $G$ with
dimension at least $2$ such that $Z_{\Gamma}(s,\rho)$ vanishes at some point
$s=\sigma+it\in R_{\eta}.$ Then, by the Fredholm determinant identity
(Proposition 2.1) and the general theory of Fredholm determinants, there
exists a non-zero 1-eigenfunction $F\in H^{2}(D,V)$ of $\mathcal{L}_{s,\rho},$
that is, we have

(82) $\mathcal{L}_{s,\rho}^{N}F=F$

for all positive integers $N$. The goal is to show that such a non-zero
function can never exist for $s\in R_{\eta}$ for some suitably small constant
$\eta=\eta(\epsilon)>0$, which we will determine later in the proof. In the
sequel, we set

(83) $f\stackrel{{\scriptstyle\text{def}}}{{=}}\varphi_{\sigma}^{-1}F$

and we let $x\in\Lambda$ be a point at which the function
$x\mapsto\|f(x)\|_{V}$ attains its maximum, i.e.,
$\|f(x)\|_{V}=\|f\|_{\infty,\Lambda}.$ (Such a point must exist because the
limit set $\Lambda\subset\mathbb{R}$ is compact.) We let $j\in[2m]$ be the
index such that $x\in\Lambda\cap I_{j}$. By Proposition 4.1 we have

(84)
$f(x)=\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))+\Delta_{\textbf{a}}$

for all words $\textbf{a}\in W_{N}^{j}$, where the norm of
$\Delta_{\textbf{a}}\in V$ is bounded from above by

(85) $\|\Delta_{\textbf{a}}\|_{V}\leq
C^{N}\sqrt{\delta-\sigma}\|f\|_{\infty,\Lambda}.$

Now we fix an orthonormal basis $\textbf{e}_{1},\dots,\textbf{e}_{d}$ for the
vector space $V$ and we define the complex conjugate of $\textbf{v}\in V$ by

$\textbf{v}=v_{1}\textbf{e}_{1}+\dots+v_{d}\textbf{e}_{d}\Longrightarrow\overline{\textbf{v}}\stackrel{{\scriptstyle\text{def}}}{{=}}\overline{v_{1}}\textbf{e}_{1}+\dots+\overline{v_{d}}\textbf{e}_{d}.$

Similarly, we define the complex conjugate of an element A in the endomorphism
ring $\mathrm{End}(V)$ by conjugating all the entries of its matrix
representation:

$\textbf{A}=\sum_{i,j}a_{ij}\textbf{e}_{i}\textbf{e}_{j}^{\ast}\Longrightarrow\overline{\textbf{A}}\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{i,j}\overline{a_{ij}}\textbf{e}_{i}\textbf{e}_{j}^{\ast}.$

It is straightforward to check the rules

(86)
$\overline{\textbf{v}+\textbf{w}}=\overline{\textbf{v}}+\overline{\textbf{w}},\quad\overline{\textbf{A}}\overline{\textbf{v}}=\overline{\textbf{A}\textbf{v}},\quad\overline{\lambda\textbf{v}}=\overline{\lambda}\overline{\textbf{v}},\quad\overline{\textbf{A}\textbf{B}}=\overline{\textbf{A}}\cdot\overline{\textbf{B}}.$

for all $\lambda\in\mathbb{C},$ for all $\textbf{v},\textbf{w}\in V,$ and for
all $\textbf{A},\textbf{B}\in\mathrm{End}(V),$ which we will use in the sequel
without explicitly referring to them.

Now observe that since the derivatives $\gamma_{\textbf{a}}^{\prime}(x)$ are
real, we have
$\overline{\gamma_{\textbf{a}}^{\prime}(x)^{it}}=\gamma_{\textbf{a}}^{\prime}(x)^{-it}.$
Thus, we can tensor multiply both sides of equation (84) with its complex
conjugate to obtain

$\displaystyle f(x)\otimes\overline{f(x)}$
$\displaystyle=\left(\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))+\Delta_{\textbf{a}}\right)\otimes\left(\overline{\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))+\Delta_{\textbf{a}}}\right)$
$\displaystyle=\left(\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))+\Delta_{\textbf{a}}\right)\otimes\left(\gamma_{\textbf{a}}^{\prime}(x)^{-it}\overline{\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))}+\overline{\Delta_{\textbf{a}}}\right)$
(87)
$\displaystyle=\left(\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\right)\otimes\left(\overline{\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))}\right)+\widetilde{\Delta}_{\textbf{a}}$

for all words $\textbf{a}\in W_{N}^{j}$, where we have set

(88)
$\widetilde{\Delta}_{\textbf{a}}\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{\textbf{a}}^{\prime}(x)^{it}\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))\otimes\overline{\Delta_{\textbf{a}}}+\Delta_{\textbf{a}}\otimes\gamma_{\textbf{a}}^{\prime}(x)^{-it}\overline{\rho(\gamma_{\textbf{a}})^{-1}f(\gamma_{\textbf{a}}(x))}+\Delta_{\textbf{a}}\otimes\overline{\Delta_{\textbf{a}}}.$

To proceed, we rewrite equation (87) more conveniently as

(89)
$h(x)=\nu(\gamma_{\textbf{a}})^{-1}h(\gamma_{\textbf{a}}(x))+\widetilde{\Delta}_{\textbf{a}},$

where

$h\stackrel{{\scriptstyle\text{def}}}{{=}}f\otimes\overline{f}$ and
$\nu\stackrel{{\scriptstyle\text{def}}}{{=}}\rho\otimes\overline{\rho}.$

Both sides of (89) are elements of the vector space
$W\stackrel{{\scriptstyle\text{def}}}{{=}}V\otimes V.$ We endow $W$ with the
natural inner product and norm which they inherit from $V.$ In particular, for
all $\textbf{v}_{1},\textbf{v}_{2}\in V,$ we have

(90)
$\|\textbf{v}_{1}\otimes\textbf{v}_{2}\|_{W}=\|\textbf{v}_{1}\|_{V}\|\textbf{v}_{2}\|_{V}.$

Thus, using the triangle inequality in $W$ and the bound in (85), we obtain
(after increasing the constant $C$):

(91) $\|\widetilde{\Delta}_{\textbf{a}}\|_{W}\leq
2\|\Delta_{\textbf{a}}\|_{V}\|f\|_{\infty,\Lambda}+\|\Delta_{\textbf{a}}\|_{V}^{2}\leq
C^{N}\sqrt{\delta-\sigma}\|f\|_{\infty,\Lambda}^{2}.$

Now we average (89) over all words $\textbf{a}\in W_{N}^{j},$ writing

(92) $h(x)=\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h(\gamma_{\textbf{a}}(x))\right)+\widetilde{\Delta},\quad\widetilde{\Delta}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\widetilde{\Delta}_{\textbf{a}}\right).$

By the triangle inequality and the bound in (91), we have

(93) $\|\widetilde{\Delta}\|_{W}\leq\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\|\widetilde{\Delta}_{\textbf{a}}\|_{W}\leq
C^{N}\sqrt{\delta-\sigma}\|f\|_{\infty,\Lambda}^{2},$

which we will use further below.

Since $(\nu,W)$ is a unitary representation of $G$, we may use Proposition 3.3
to estimate the average on the right hand side of (92), except that the
function $h$ may not satisfy the necessary assumption in (55). Thus we
decompose $h$ as

(94) $h=h_{1}+h_{2},$

where for all $x\in\Lambda,$

$h_{1}(x)=\frac{1}{|G|}\sum_{g\in G}\nu(g)h(x)$

is the orthogonal projection of $h(x)$ onto the trivial subrepresentation in
$W$, and $h_{2}=h-h_{1}.$

######  Lemma 4.4.

Notations and assumptions being as above, we have

$\|h_{1}\|_{\infty,\Lambda}\leq\frac{1}{\sqrt{\dim(\rho)}}\|f\|_{\infty,\Lambda}^{2}.$

Before we prove this lemma, let us finish the proof of the main theorem in the
case $\dim(\rho)\geq 2$. In view of the decomposition in (94), we write

$h(x)=S_{1}+S_{2}+\widetilde{\Delta},$

where

$S_{i}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left(\nu(\gamma_{\textbf{a}})^{-1}h_{i}(\gamma_{\textbf{a}}(x))\right),\quad
i=1,2.$

Using the triangle inequality, we get

(95)
$\|h(x)\|_{W}\leq\left\|S_{1}\right\|_{W}+\left\|S_{2}\right\|_{W}+\|\widetilde{\Delta}\|_{W}.$

We will now estimate the three terms on the right individually. Using the
triangle inequality again, the unitary of $\nu$ with respect to the norm
$\|\cdot\|_{W}$, and Lemma 4.4, we get

(96) $\left\|S_{1}\right\|_{W}\leq\mathbb{E}_{\textbf{a}\in
W_{N}^{j}}\left\|h_{1}(\gamma_{\textbf{a}}(x))\right\|_{W}\leq\|h_{1}\|_{\infty,\Lambda}\leq\frac{1}{\sqrt{\dim(\rho)}}\|f\|_{\infty,\Lambda}^{2}.$

We can now exploit the assumption that $\rho$ has dimension at least $2$ to
estimate

$\frac{1}{\sqrt{\dim(\rho)}}\leq\frac{1}{\sqrt{2}}<0.8,$

say, which when inserted into (96) yields

(97) $\left\|S_{1}\right\|_{W}\leq 0.8\|h\|_{\infty,\Lambda}.$

Let us now turn our attention to the second term. By construction, $h_{2}$
satisfies

$\sum_{g\in G}\nu(g)h_{2}(x)=0$

for all $x\in\Lambda$, so we can use Proposition 3.3 to obtain

(98) $\left\|S_{2}\right\|_{W}\ll e^{-c\epsilon
N}\left(\|h_{2}\|_{\infty,\Lambda}+[h_{2}]_{\Lambda}\right).$

Recall that $h=f\otimes\overline{f},$ so

$\|h_{2}\|_{\infty,\Lambda}\leq\|h\|_{\infty,\Lambda}=\|f\|_{\infty,\Lambda}^{2}.$

Moreover, by the definition of the semi-norm $[\cdot]_{\Lambda}$ in (54), we
have

$[h_{2}]_{\Lambda}\leq[h]_{\Lambda}=[f\otimes\overline{f}]_{\Lambda}\ll\|f\|_{\infty,\Lambda}[f]_{\Lambda}.$

By Lemma 4.3, the semi-norm $[f]$ can be estimated from above as

$[f]_{\Lambda}\ll\|f\|_{\infty,\Lambda},$

so we obtain furthermore

$[h_{2}]_{\Lambda}\ll\|f\|_{\infty,\Lambda}^{2}.$

Thus, returning to (98), we obtain

(99) $\left\|S_{2}\right\|_{W}\ll e^{-c\epsilon N}\|f\|_{\infty,\Lambda}^{2}.$

Gathering the estimates (97), (99) and (93), and inserting them into (95), we
find

$\|f\|_{\infty,\Lambda}^{2}\leq 0.8\|f\|_{\infty,\Lambda}^{2}+Ce^{-c\epsilon
N}\|f\|_{\infty,\Lambda}^{2}+C^{N}\sqrt{\delta-\sigma}\|f\|_{\infty,\Lambda}^{2}$

for some well-chosen constant $C$ depending only on $\Gamma.$ Since $F$ is
non-zero, so is $f$. Thus we can divide both sides by
$\|f\|_{\infty,\Lambda}^{2}\neq 0$ to get

(100) $1\leq 0.8+Ce^{-c\epsilon N}+C^{N}\sqrt{\delta-\sigma}.$

To achieve the desired contradiction to (82), we choose

(101) $N=\left\lceil c^{-1}\epsilon^{-1}\log(10C+1)\right\rceil$

so that

(102) $Ce^{-c\epsilon N}<0.1$

and we choose

(103) $\eta=0.008C^{-2N}$

so that for all $\sigma\geq\delta-\eta$, we have

(104) $C^{N}\sqrt{\delta-\sigma}\leq C^{N}\sqrt{\eta}<0.09$

The choice of $N$ and $\eta$ yields

$1<0.8+0.1+0.09=0.99,$

a contradiction. Thus the equation $\mathcal{L}_{s,\rho}^{N}F=F$ can not hold
true for non-zero $F$ for $s\in R_{\eta}$ with $\eta$ as in (103).
Consequently, $Z_{\Gamma}(s,\rho)$ does not vanish in $s\in R_{\eta}$ for all
irreducible representations $\rho$ of $G$ whose dimension is at least two. To
conclude, notice that by inserting (101) into (103), we obtain

$\eta=c_{1}\exp\left(-c_{2}\epsilon^{-1}\right)$

for some positive constants $c_{1}$ and $c_{2}$ depending only on $\Gamma.$

It remains to prove Lemma 4.4.

###### Proof of Lemma 4.4.

Recall that $(\rho,V)$ is an irreducible representation of $G,$
$h=f\otimes\overline{f}\in W=V\otimes V$ and $\nu=\rho\otimes\overline{\rho}.$
Thus for all $x\in\Lambda$ we can write

(105) $h_{1}(x)=\frac{1}{|G|}\sum_{g\in G}\nu(g)h(x)=\frac{1}{|G|}\sum_{g\in
G}\rho(g)f(x)\otimes\overline{\rho(g)f(x)}.$

Consider the linear bijection

(106) $\mathsf{m}\colon W=V\otimes
V\to\mathrm{End}(V),\quad\sum_{i,j}a_{ij}\textbf{e}_{j}\otimes\textbf{e}_{j}\mapsto\sum_{i,j}a_{ij}\textbf{e}_{j}\cdot\textbf{e}_{j}^{\ast},$

where $\textbf{A}^{\ast}$ denotes the adjoint of $\textbf{A},$ and observe
that for all $\textbf{w}\in W,$

(107) $\|\textbf{w}\|_{W}=\|\mathsf{m}(\textbf{w})\|_{2},$

where
$\|\textbf{A}\|_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\sqrt{\mathrm{tr}(\textbf{A}^{\ast}\textbf{A})}$
is the Frobenius norm. Notice also that for elements of the form
$\textbf{v}_{1}\otimes\textbf{v}_{2}$ with $\textbf{v}_{1},\textbf{v}_{2}\in
V,$ the map $\mathsf{m}$ takes the form

$\mathsf{m}(\textbf{v}_{1}\otimes\textbf{v}_{2})=\textbf{v}_{1}\cdot\textbf{v}_{2}^{\top},$

where
$\textbf{v}^{\top}\stackrel{{\scriptstyle\text{def}}}{{=}}\overline{\textbf{v}}^{\ast}$
denotes the transpose of $\textbf{v}.$ Hence

$\displaystyle\mathsf{m}(h_{1}(x))$
$\displaystyle=\mathsf{m}\left(\frac{1}{|G|}\sum_{g\in
G}\rho(g)f(x)\otimes\overline{\rho(g)f(x)}\right)$
$\displaystyle=\frac{1}{|G|}\sum_{g\in
G}\mathsf{m}\left(\rho(g)f(x)\otimes\overline{\rho(g)f(x)}\right)$
$\displaystyle=\frac{1}{|G|}\sum_{g\in
G}\rho(g)f(x)\left(\overline{\rho(g)f(x)}\right)^{\top}$
$\displaystyle=\frac{1}{|G|}\sum_{g\in
G}\rho(g)f(x)\left(\rho(g)f(x)\right)^{\ast}$
$\displaystyle=\frac{1}{|G|}\sum_{g\in
G}\rho(g)f(x)f(x)^{\ast}\rho(g)^{\ast}.$

Since $\rho$ is unitary, we have $\rho(g)^{\ast}=\rho(g)^{-1}$ for all $g\in
G,$ whence

(108) $\mathsf{m}(h_{1}(x))=\frac{1}{|G|}\sum_{g\in
G}\rho(g)f(x)f(x)^{\ast}\rho(g)^{-1}.$

The previous equation shows that $\mathsf{m}(h_{1}(x))$ is a self-intertwiner,
that is, we have

$\rho(h)\mathsf{m}(h_{1}(x))\rho(h)^{-1}=\mathsf{m}(h_{1}(x))$

for all $h\in G.$ Since $\rho$ is irreducible by assumption, we conclude from
Schur’s Lemma that $\mathsf{m}(h_{1}(x))$ is a multiple of the identity, that
is, there exists some $\lambda\in\mathbb{C}$ such that

$\mathsf{m}(h_{1}(x))=\lambda I_{V}.$

Taking traces on both sides reveals that

$\lambda=\frac{\mathrm{tr}_{V}(f(x)f(x)^{\ast})}{\dim(\rho)}=\frac{\|f(x)\|_{V}^{2}}{\dim(\rho)}.$

Finally, using (107), we obtain

$\|h_{1}(x)\|_{W}=\lambda\|I_{V}\|_{2}=\lambda\sqrt{\dim(\rho)}=\frac{\|f(x)\|_{V}^{2}}{\sqrt{\dim(\rho)}},$

which concludes the proof. ∎

#### The case $\dim(\rho)=1$

Now suppose that the group $G$ has a non-trivial, one-dimensional
representation $\rho\colon G\to\mathbb{C}^{\times}$ such that
$Z_{\Gamma}(s,\rho)$ vanishes at some point $s\in R_{\eta},$ where $R_{\eta}$
is the rectangular box in (81). The one-dimensional case requires special
treatment, since the argument in the previous paragraph does not go through in
(97), for it is crucial to have an absolute factor strictly less than $1$
(such as $0.8$) on the right hand side of (97); replacing it by $1$ would
destroy the argument.

First, observe that since $\Gamma$ is a free group on $m$ generators, we have

(109)
$H^{1}(\Gamma,\mathbb{Z})\stackrel{{\scriptstyle\text{def}}}{{=}}\Gamma/[\Gamma,\Gamma]\cong\mathbb{Z}^{m}.$

Hence, all one-dimensional representations (that is, characters) of $\Gamma$
are parametrized by the characters of the group $\mathbb{Z}^{m}.$ More
precisely, if $P\colon\Gamma\to\mathbb{Z}^{m}$ is a fixed surjective
homomorphism realizing (109), all the characters of $\Gamma$ have the form

$\chi_{\theta}(\gamma)\stackrel{{\scriptstyle\text{def}}}{{=}}e^{2\pi
i\langle\theta,P(\gamma)\rangle},$

where $\theta=(\theta_{1},\dots,\theta_{m})$ belongs to the $m$-dimensional
torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\mathbb{Z}^{m}$ and

$\langle\theta,x\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}\sum_{j=1}^{m}\theta_{j}x_{j},\quad
x=(x_{1},\dots,x_{m})$

is the usual inner product. Given a character $\chi_{\theta},$ we write

$Z_{\Gamma}(s,\theta)\stackrel{{\scriptstyle\text{def}}}{{=}}Z_{\Gamma}(s,\chi_{\theta})$

for notational convenience. Since the extension of $\rho$ to $\Gamma$ is also
a character, there exists $\theta_{0}\in\mathbb{T}^{m}$ such that

(110) $Z_{\Gamma}(s,\rho)=Z_{\Gamma}(s,\theta_{0}).$

Let $\|\theta\|=\sqrt{\theta_{1}^{2}+\dots\theta_{m}^{2}}$ be the euclidean
norm. From [21, Proposition 4.2] we know that

$Z_{\Gamma}(\delta+it,\theta)=0\Longleftrightarrow(t,\theta)=(0,\textbf{0}).$

By a compactness argument (for the details we refer to [21, Section 4.1]), we
can then deduce the following statement: for all $\widetilde{\epsilon}>0$
there exists $\widetilde{\eta}=\widetilde{\eta}(\widetilde{\epsilon})>0$ such
that if $\theta\in\mathbb{T}^{m}$ satisfies
$\|\theta\|\geq\widetilde{\epsilon},$ then $Z_{\Gamma}(s,\theta)$ does not
vanish inside the rectangle $R_{\widetilde{\eta}}.$ Hence, it suffices to show
that we have $\|\theta_{0}\|\geq\widetilde{\epsilon}$ for some constant
$\widetilde{\epsilon}=\widetilde{\epsilon}(\epsilon)>0$ depending on the
expansion constant $\epsilon.$

We now claim that $\|\theta_{0}\|\geq c\sqrt{\epsilon}$ for some constant
$c>0.$ To prove this, suppose for a contradiction that
$\|\theta_{0}\|<c\sqrt{\epsilon}.$ Then, by the Cauchy–Schwarz inequality, we
have

$|\langle\theta_{0},P(\gamma_{j})\rangle|\leq\|\theta_{0}\|\|P(\gamma_{j})\|<cK\sqrt{\epsilon}$

for all $j\in[2m],$ where

$K\stackrel{{\scriptstyle\text{def}}}{{=}}\max_{j\in[2m]}\|P(\gamma_{j})\|.$

Moreover, using the elementary inequality $\cos(2\pi x)>1-2\pi^{2}x^{2}$ for
$|x|$ small, we obtain

(111)
$\frac{1}{2m}\sum_{j=1}^{2m}\chi_{\theta_{0}}(\gamma_{j})=\frac{1}{m}\sum_{j=1}^{m}\cos\left(2\pi\langle\theta,P(\gamma_{j})\rangle\right)>1-2\pi^{2}c^{2}K^{2}\epsilon.$

But on the other hand, with the same reasoning leading to (36), we find that

(112)
$\left|\frac{1}{2m}\sum_{j=1}^{2m}\chi_{\theta_{0}}(\gamma_{j})\right|\leq
1-\epsilon.$

It is now clear that for $c$ small enough, (111) contradicts (112). This
finishes the proof.

## References

  * [1] A. Beardon, _The exponent of convergence of Poincaré series_ , Proc. Lond. Math. Soc. (3) 18 (1968), 461–483.
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