Explicit spectral gaps for random covers of Riemann surfaces

TL;DR

This paper establishes probabilistic bounds on the absence of new resonances and eigenvalues in random covers of hyperbolic surfaces, providing explicit spectral gap estimates with high probability as the cover degree grows.

Contribution

It introduces a permutation model for random covers of hyperbolic surfaces and proves new spectral gap results for resonances and eigenvalues in these covers.

Findings

No new resonances in a specified strip with high probability

Explicit spectral gap for new eigenvalues when Hausdorff dimension exceeds 1/2

High probability absence of new resonances near the critical line

Abstract

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$, and similarly define new eigenvalues of the Laplacian. We prove that for any $\epsilon>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=\sigma+it$ of $X_{n}$ with $\sigma\in[\frac{3}{4}\delta+\epsilon,\delta]$ and $t\in[-H,H]$. This implies in the case of $\delta>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic `high frequency' resonance-free strip result, we obtain the corollary that there is an $\eta=\eta(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>\delta-\eta\,\}$.

Equations

$$k={\lambda }_0\ge {\lambda }_1\ge \cdots \ge {\lambda }_n$$

$$\gamma .(x,i)=(\gamma (x),{\varphi }_n(\gamma )(i)).$$

$$X_n=\underset{i\in S}{⨆}{\Gamma }_i\backslash \mathbb{H}.$$

$$\sum _{i\in S}[\Gamma :{\Gamma }_i]=n.$$

$$\mathrm{spec}({\Delta }_{X_n})\cap \left[\delta \left(1-\delta \right),{\sigma }_0(1-{\sigma }_0)\right]=\mathrm{spec}({\Delta }_X)\cap \left[\delta \left(1-\delta \right),{\sigma }_0(1-{\sigma }_0)\right]$$

$$\mathrm{spec}({\Delta }_{X_n})\cap (\delta \left(1-\delta \right),{\sigma }_0(1-{\sigma }_0)]=\varnothing ,$$

$$R_Y(s)\stackrel{\mathrm{def}}{=}({\Delta }_Y-s(1-s){)}^{-1}:L^2(Y)\to L^2(Y)$$

$${\mathcal{R}}_Y\cap \{s:\mathrm{Re}(s)\ge \delta -{ϵ}_{\Gamma }\}=\{\delta \}.$$

$${\mathcal{R}}_Y\cap \{s:\mathrm{Re}(s)\ge \frac{1}{2}-\widetilde{ϵ}\}$$

$$\mathrm{Rect}({\sigma }_0,H)\stackrel{\mathrm{def}}{=}\{s=\sigma +it:\sigma \in [{\sigma }_0,\delta ]\mathrm{and}|t|\le H\}.$$

$${\mathcal{R}}_{X_n}\bigcap \mathrm{Rect}({\sigma }_0,H)={\mathcal{R}}_X\bigcap \mathrm{Rect}({\sigma }_0,H)$$

$${\mathcal{R}}_{\tilde{X}}\cap \{s:\mathrm{Re}(s)\ge \delta -{ϵ}_{\Gamma }\mathrm{and}|\mathrm{Im}(s)|\ge T_{\Gamma }\}=\varnothing .$$

$${\mathcal{R}}_X\cap \{\mathrm{Re}(s)\ge \delta -\epsilon (\delta )\}$$

$${\lambda }_1(Y^{\prime })\ge c(Y)\left({\lambda }_0\left(G\right)-{\lambda }_1(G)\right).$$

$${\mathbb{E}}_n[\mathrm{Tr}({\rho }_n^0({\gamma }_{{\mathbf{a}}^{\prime }}{\gamma }_{{\mathbf{b}}^{\prime }}^{-1}))]$$

$$\sum _{\mathbf{a},\mathbf{b}\in \overline{Z}(\tau )}|{\mathbb{E}}_n[\mathrm{Tr}({\rho }_n^0({\gamma }_{{\mathbf{a}}^{\prime }}{\gamma }_{{\mathbf{b}}^{\prime }}^{-1}))]|$$

$$\sum _{(\mathbf{a},\mathbf{b})\in \mathrm{PowerPairs}(\tau )}{{\rm Y}}_{\mathbf{a}}^{\frac{\delta }{2}}{{\rm Y}}_{\mathbf{b}}^{\frac{\delta }{2}};$$

$${\gamma }_a\left(\hat{\mathbf{C}}-D_{\bar{a}}\right)=\overline{D_a},{\gamma }_{\bar{a}}={\gamma }_a^{-1}.$$

$$\mathbf{a}=(a_1,\dots ,a_n),n\in \mathbf{N}\cup \{0\}$$

$$D_{\mathbf{a}}={\gamma }_{{\mathbf{a}}^{\prime }}(D_{a_n}),I_{\mathbf{a}}={\gamma }_{{\mathbf{a}}^{\prime }}(I_{a_n})$$

$$T{|}_{D_a}={\gamma }_a^{-1}={\gamma }_{\bar{a}}.$$

$$\Lambda (\Gamma )=\bigcap _{n=1}^{\infty }{T}^{-n}(\mathbf{D}).$$

$$P(\phi )=\underset{\mu }{\sup }\left(h_{\mu }(T)+{\int }_{\Lambda }\phi d\mu \right),$$

$$\sigma \mapsto P(-\sigma \log |T^{\prime }|)$$

$$⟨f,g⟩\stackrel{\mathrm{def}}{=}{\int }_{\Omega }⟨f(x),g(x){⟩}_{V}dm(x).$$

$$\mathcal{H}(\mathbf{D};V)=\underset{a\in \mathcal{A}}{⨁}\mathcal{H}(D_a;V).$$

$$\sum _{k=1}^{\infty }{\mathbf{e}}_k(x_1)\overline{{\mathbf{e}}_k(x_2)}\stackrel{\mathrm{def}}{=}{B}_{D_a}(x_1,x_2)$$

$$B_{D_a}(x_1,x_2)=\frac{r_a^2}{\pi {\left[r_a^2-({\overline{x}}_2-c_a)(x_1-c_a)\right]}^2}$$

$${\mathcal{L}}_{Z,s,\rho }[f](x)$$

$${\mathcal{L}}_{s,\rho }[f](x)=\sum _{\begin{array}{c}a\in \mathcal{A}\\ a\to b\end{array}}{\gamma }_a^{\prime }(x{)}^s\rho ({\gamma }_a^{-1})f({\gamma }_a(x))x\in D_b,b\in \mathcal{A}.$$

Full text

We prove that for any $\epsilon>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=\sigma+it$ of $X_{n}$ with $\sigma\in[\frac{3}{4}\delta+\epsilon,\delta]$ and $t\in[-H,H]$. This implies in the case of $\delta>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic ‘high frequency’ resonance-free strip result, we obtain the corollary that there is an $\eta=\eta(X)$ such that with probability $\to 1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>\delta-\eta\,\}$.

1. 1 Introduction

2. 1.1 Prior work

3. 1.2 Overview of proofs and paper organization

4. 1.3 Notation

5. 1.4 Acknowledgments

6. 2 Preliminaries

7. 2.1 Words, encodings of Schottky groups, and pressure

8. 2.2 Functional spaces and transfer operators

9. 2.3 The representations appearing in this paper

10. 2.4 Selberg zeta functions

11. 3 Estimates for derivatives

12. 4 Transfer operators and zeta functions

13. 4.1 Zeta functions

14. 4.2 The Hilbert-Schmidt norm of the transfer operator

15. 4.3 A pointwise estimate for the modulus of a zeta function

16. 5 The expectation of the Hilbert-Schmidt norm of the transfer operator

17. 5.1 Statement of the main probabilistic estimate

18. 5.2 The expected value of the trace of a word

19. 5.3 Majorization of the expectation of the Hilbert-Schmidt norm

20. 5.4 Estimating the size of $\mathrm{PowerPairs}$($\tau$)

21. 5.5 Proof of Proposition 5.1

22. 6 Proof of Theorems 1.1 and 1.8

23. 7 Proof of Theorem 1.10 about high frequency resonances

1. Introduction

This paper is about spectral gaps for random Riemann surfaces. More specifically, we are interested in various notions of spectral gap for random covers of a fixed Schottky Riemann surface. This is in close analogy to questions about the spectral gap of a random regular graph, and this analogy informs our model for random coverings, so we begin with a discussion on graphs.

Let $G$ be a $k$-regular graph on $n$ vertices. Then the adjacency matrix $A_{G}$ of $G$ has $n$ real eigenvalues in $[-k,k]$ and $k$ appears as an eigenvalue with multiplicity equal to the number of connected components of $G$. Denoting by

[TABLE]

the eigenvalues of $G$, the spectral gap of $G$ is $\lambda_{0}-\lambda_{1}$. If $G$ is connected, then $\lambda_{0}>\lambda_{1}$ and the spectral gap is related to the exponential rate at which the random walk on $G$ converges to the uniform measure. As such, it is an important quantity in theoretical computer science, and accordingly, there has been a great deal of interest in the spectral gap of a random regular graph. Alon’s conjecture [Alo86], now a theorem due to Friedman [Fri08], says that for any $\epsilon>0$, as $n\to\infty$, the probability that $\lambda_{1}(G_{n})>2\sqrt{k-1}+\epsilon$ tends to zero, when $G_{n}$ is sampled uniformly at random from $k$-regular graphs with $n$ vertices. The relevance of the quantity $2\sqrt{k-1}$ is that for any $k$-regular graph with $n$ vertices, a result of Alon-Boppana [Nil91] says that $\lambda_{1}(G)\geq 2\sqrt{k-1}-o_{n}(1)$, so $2\sqrt{k-1}$ is an asymptotically optimal lower bound for $\lambda_{1}(G)$, often called the Ramanujan bound after [LPS88].

The model of a random graph described above chooses random graphs according to a uniform distribution. Another popular model for a random $2k$-regular graph is called the *permutation model *and is the one we wish to focus on in the sequel. Let $\Gamma=\langle\gamma_{1},\ldots,\gamma_{k},\gamma_{1}^{-1},\ldots,\gamma_{k}^{-1}\rangle$ be a free group on $k$ generators, $k\geq 2$, and let $S_{n}$ denote the symmetric group on $n$ letters, and $\phi_{n}$ be a random homomorphism from $\Gamma$ to $S_{n}$, sampled uniformly from all possible homomorphisms. Since $\Gamma$ is free, a homomorphism is described simply by choosing the images $\phi_{n}(\gamma_{i})$ of the generators of $\Gamma$ independently and uniformly from $S_{n}$. Then let $G_{n}$ be the random graph with vertex set $[n]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{1,\ldots,n\}$ and with an edge between $i$ and $j$ if there is a generator $\gamma_{a}$ such that $\phi_{n}(\gamma_{a})(i)=j$. We will adapt this model to a model of a random Riemann surface.

Let $X$ be a connected, non-elementary, non-compact, convex co-compact hyperbolic surface. Then $X=\Gamma\backslash\mathbb{H}$ where $\mathbb{H}$ is the hyperbolic upper half plane and $\Gamma$ is a free subgroup of $\mathrm{SL}_{2}(\mathbf{R})$. We view $X$ as fixed throughout the paper. We let $X_{n}$ be the random $n$-cover of $X$ obtained as a fibered product $X_{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathbb{H}\times_{\phi_{n}}[n]$. More precisely, $X_{n}$ is the quotient of $\mathbb{H}\times[n]$ by the diagonal action of $\Gamma$

[TABLE]

If $S\subset[n]$ is a set of representatives for the orbits of $\Gamma$ on $[n]$ via $\phi_{n}$, and $\Gamma_{i}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathrm{Stab}_{\Gamma}(i)$ is the stabilizer of $i\in S$, then $X_{n}$ is isomorphic to the disjoint union of (connected) covers $\Gamma_{i}\backslash\mathbb{H}$, i.e.

[TABLE]

Notice that we have

[TABLE]

We say that a property $E(\phi_{n})$ of the random $\phi_{n}$ holds asymptotically almost surely (a.a.s.) if as $n\to\infty$, the probability that $E(\phi_{n})$ holds tends to $1$. It is an elementary calculation111For a concrete reference, this statement follows from [BS87, Thm.13]. One can also prove by elementary combinatorial arguments that the probability that $X_{n}$ is connected as $n\rightarrow+\infty$ is bigger than $1-\frac{C(k)}{n^{k-1}}$, where $C(k)>0$ is a constant depending only on $k$. that a.a.s. $\Gamma$ acts transitively on $[n]$ via $\phi_{n}$ and hence, a.a.s. $X_{n}$ is connected. This also follows from the main theorems below. Although we do not assume $X_{n}$ is connected at any point, it would not hurt to assume this on a first reading.

We now discuss the spectral theory of $X$ and $X_{n}$. The group $\Gamma$ acts properly discontinuously on $\mathbb{H},$ but for any point $o\in\mathbb{H}$, the orbit $\Gamma o$ accumulates on $\partial\mathbb{H}=\mathbf{R}\cup\{\infty\}$ and the accumulation set of this orbit is called the *limit set of $\Gamma$ *and denoted by $\Lambda(\Gamma)$. This $\Lambda(\Gamma)$ is a perfect nowhere dense fractal and has an associated Hausdorff dimension $\delta\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\dim_{\mathrm{Haus}}(\Lambda(\Gamma))\in[0,1)$. By a result of Lax and Phillips [LP81], the spectrum of the Laplacian $\Delta_{X}$ is discrete in the range $[0,\frac{1}{4})$, and Patterson [Pat76] proved that if $\delta>\frac{1}{2}$, then the lowest eigenvalue of $\Delta_{X}$ is $\delta(1-\delta)$. If $\delta\leq\frac{1}{2}$ then there are no eigenvalues of $\Delta_{X}$. The same is true for $X_{n}$, with the same $\delta$ (although $\delta(1-\delta)$ will be simple if and only if $X_{n}$ is connected). More generally, if $\lambda$ is any eigenvalue of $X$, then by lifting eigenfunctions through the covering map, $\lambda$ is an eigenvalue for $X_{n}$ with at least as large multiplicity.

The first main theorem of our paper is the following.

Theorem 1.1**.**

Assume that $\delta>\frac{1}{2}$. Then for any $\sigma_{0}\in\left(\frac{3}{4}\delta,\delta\right)$, a.a.s.

[TABLE]

and the multiplicities on both sides are the same.

Remark 1.2*.*

This theorem implies that a.a.s. the $X_{n}$ have a uniform spectral gap, and this spectral gap only depends on $\delta$ and the gap between the first two eigenvalues of $X$.

Remark 1.3*.*

If $\delta\in(\frac{1}{2},\frac{2}{3})$ then since $X_{n}$ has no eigenvalues in $[\frac{1}{4},\infty)$ by a result of Lax and Phillips [LP81], Theorem 1.1 implies that a.a.s. $X_{n}$ has no new eigenvalues.

Remark 1.4*.*

Theorem 1.1 can be viewed as a significant sharpening of a result of Brooks and Makover [BM04], albeit in the infinite area setting. See $\S$1.1 for a more detailed discussion of this comparison.

Remark 1.5*.*

We point out that it is possible for $X_{n}$ to not be connected, and in this case, there is no spectral gap. Even further, it is easy to see that $X_{n}$ can be a connected cyclic cover of $X$, and by results of [JNS19], these have no uniform spectral gap.

Remark 1.6*.*

In the limit as $\delta\to 1$, the range of forbidden eigenvalues in (1.1) becomes $[0,\frac{3}{16})$. This is interestingly the same range covered by Selberg’s $\frac{3}{16}$ Theorem [Sel65] on the spectral gap of congruence covers of the modular surface $\mathrm{SL}_{2}(\mathbf{Z})\backslash\mathbb{H}.$ This should also be compared to the deterministic result of Gamburd [Gam02] for congruence covers of infinite index geometrically finite subgroups of $\mathrm{SL}_{2}(\mathbf{Z})$: assuming $\delta>\frac{5}{6}$, he shows that the spectrum remains the same in the range $[\delta(1-\delta),\frac{5}{36})$. See also [Mag15] for a generalization of this result to higher dimensions.

We write $\chi(X)$ for the Euler characteristic of $X$. It has recently been proved by Ballmann, Matthiesen, and Mondal [BMM17] that if $\chi(X)<0$, $\Delta_{X}$ has at most $-\chi(X)$ eigenvalues. If $\chi(X)=-1$ then this means the only possible eigenvalue of $X$ is at $\delta(1-\delta)$ and thus Theorem 1.1 yields

Corollary 1.7**.**

Assume that $\delta>\frac{1}{2}$. If $X$ is topologically a pair of pants, or a torus with one hole, then for any $\sigma_{0}\in\left(\frac{3}{4}\delta,\delta\right)$, a.a.s.

[TABLE]

and $\delta(1-\delta)$ is a simple eigenvalue of $\Delta_{X_{n}}$.

We now turn to what we can say about general $\delta\in(0,1)$. In the case $\delta\leq\frac{1}{2}$, $\Delta_{X}$ and $\Delta_{X_{n}}$ will have no discrete $L^{2}$ spectrum, so one must consider a more subtle notion of spectral gap.

For any non-elementary convex co-compact hyperbolic $Y$ with $\delta=\delta(Y)$ (e.g. $Y=X$, $Y=X_{n}$) the resolvent

[TABLE]

is, a priori, a meromorphic family of bounded operators in the right half plane $\mathrm{Re}(s)>\frac{1}{2}$ with poles precisely at real $s$ such that $s(1-s)$ is an eigenvalue of $\Delta_{X}.$ By work of Mazzeo-Melrose [MM87], it can be meromorphically continued to a family of bounded operators from $C_{0}^{\infty}(Y)\rightarrow C^{\infty}(Y)$ that is meromorphic in $s\in\mathbf{C}$. In the case of hyperbolic surfaces, a simpler proof of the meromorphic continuation is due to Guillopé and Zworski [GZ95], see also the book [Bor16].

The poles of the meromorphically continued resolvent are called resonances of $Y$**. In the sequel we write $\mathcal{R}_{Y}\subset\mathbf{C}$ for the multi-set of resonances, repeated according to multiplicities222Following [Bor16, Def. 8.2], the multiplicity of a resonance $s$ of $Y$ is given by $\mathrm{rank}\left(\int_{\gamma}R_{Y}(s)ds\right)$ where $\gamma$ is an anticlockwise oriented circle enclosing $s$ and no other resonance of $Y$. . Resonances, unlike $L^{2}$-eigenvalues, correspond to a non self-adjoint spectral problem and are therefore notoriously difficult to study. There is however a clear analog of the spectral gap in this setting. The ‘bass resonance’ is located at $s=\delta$ and by a result of Naud [Nau05a] if $Y$ is connected then there exists a constant $\epsilon_{\Gamma}>0$ such that

[TABLE]

We call the existence of such a resonance free strip a spectral gap for $Y$. The spectral gap on hyperbolic surfaces has numerous applications, from prime geodesic theorems [Nau05b] to local $L^{2}$-asymptotics of waves [GN09]. A recent breakthrough of Bourgain-Dyatlov [BD18] showed that there always exists an “essential spectral gap” past the line $\{\mathrm{Re}(s)=\frac{1}{2}\}$, i.e. there exists $\tilde{\epsilon}=\tilde{\epsilon}(Y)>0$ such that

[TABLE]

is a finite set. The proof is based on the general phenomenon of “fractal uncertainty principle”, see [Dya19]. We point out that $\widetilde{\epsilon}>0$ can be made explicit, see Jin-Zhang [JZ17] and also Dyatlov-Jin [DJ18]. For a broader view and a state of the art survey on the mathematical theory of resonances including hyperbolic manifolds and related conjectures, we recommend to read [Zwo17]. Our next main result is the following.

Theorem 1.8**.**

Fix any $H>0$ and $\sigma_{0}\in\left(\frac{3}{4}\delta,\delta\right)$, and let

[TABLE]

Then a.a.s.

[TABLE]

where the multiplicities on both sides are the same.

Remark 1.9*.*

Because all eigenvalues $\lambda_{\sigma}=\sigma(1-\sigma)$ of $\Delta_{X_{n}}$ with $\sigma>\frac{1}{2}$ give a resonance of $X_{n}$ at $\sigma$, with the same multiplicity, and the same is true for $X$, Theorem 1.8 implies Theorem 1.1 and extends it to resonances in rectangles of explicit width and any bounded height $2H$. We point out that Theorem 1.8 actually yields new information on low frequency resonances past the line $\{\mathrm{Re}(s)=\frac{1}{2}\}$ when $\delta\in(\frac{1}{2},\frac{2}{3})$.

This leaves the question of how to deal with resonances with large imaginary part. For this we have the following theorem that applies to arbitrary covers. Note that here there is no randomness involved.

Theorem 1.10**.**

Assume that $\Gamma$ is a non-elementary convex co-compact group. Then there exist $\epsilon_{\Gamma}>0$ and $T_{\Gamma}>0$ such that for all finite index subgroups $\widetilde{\Gamma}\subset\Gamma$, we have for $\widetilde{X}=\widetilde{\Gamma}\backslash\mathbb{H}$,

[TABLE]

Remark 1.11*.*

From the work of Bourgain and Dyatlov [BD17], we know that there exists $\varepsilon(\delta)>0$, depending only on $\delta$ and thus uniform on covers such that

[TABLE]

is a finite set. However the result of Bourgain and Dyatlov does not provide any information on the finite set of resonances in this uniform strip. *Theorem 1.10 *shows that new resonances can only appear in a compact region.

Combining Theorem 1.8 with Theorem 1.10 yields the following corollary.

Corollary 1.12**.**

A.a.s. the random cover $X_{n}\rightarrow X$ has a uniform spectral gap. In particular, above each non elementary surface $X$, one can produce an infinite family of covers $X_{n}$ with degree $n$ and having a uniform spectral gap.

Remark 1.13*.*

When $\delta>\frac{1}{2}$, Corollary 1.12 follows from a mild extension of [BGS11, Thm. 1.2] together with results on random graphs as explained in $\S$1.1. However, when $\delta\leq\frac{1}{2}$, to our knowledge, Corollary 1.12 is completely new: the only result of that type so far is for congruence covers of convex co-compact subgroups of $\mathrm{SL}_{2}(\mathbf{Z})$, see Oh-Winter [OW16] and the discussion below.

1.1. Prior work

**Brooks and Makover. **Brooks and Makover in [BM04] consider a similar model for random finite area Riemann surfaces. In this model, random surfaces are modeled by random 3-regular oriented graphs sampled according to a refinement of the Bollobás ‘bin model’ introduced in [Bol88]. Then Brooks and Makover [BM04] construct from a random oriented graph on $n$ vertices a Riemann surface $Y_{n}$, tiled by a specific hyperbolic triangle with one vertex at $\infty$. They then consider a compactification $Y_{n}^{c}$ of the cusped surface $Y_{n}$. Thus $Y_{n}^{c}$ is a random compact Riemann surface; the genus of $Y_{n}^{c}$ is however not deterministic333By a result of Gamburd [Gam06], if $l(Y_{n})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{n}{2}+2-2\mathrm{genus}(Y_{n}^{c})$, then as $n\to\infty$, $l(Y_{n})$ converges to a Poisson-Dirichlet distribution. The function $l(Y_{n})$ coincides with the number of cusps of $Y_{n}$.. Brooks and Makover proved in (ibid.)

Theorem 1.14** (Brooks-Makover).**

There is some constant $C>0$ such that a.a.s. the first non-zero eigenvalue of $Y_{n}^{c}$ is $\geq C$.

Although our main theorems deal instead with infinite area Riemann surfaces, they offer two improvements over Theorem 1.14:

• •

The range of new forbidden eigenvalues and resonances in Theorems 1.1 and 1.8 are explicit,

• •

Moreover, we have an entire moduli space of random families (parameterized by the modulus of $X$) and the range of forbidden eigenvalues and resonances only depends on $X$ in a very mild way, through the Hausdorff dimension of the limit set.

*The Brooks-Burger transfer principle. ***Also relevant to the current work is the following transfer principle for small eigenvalues developed independently by Brooks and Burger in [Bro86, Bur88].

Theorem 1.15** (Brooks-Burger).**

Let $Y$ be any compact Riemannian manifold with $\Gamma=\pi_{1}(Y)$. There is a constant $c(Y)>0$ and a finite subset $S\subset\Gamma$ such that the following hold. Let $\Gamma^{\prime}$ be any finite index subgroup of $\Gamma$, with associated Riemannian covering space $Y^{\prime}$ of $Y$. Let $\lambda_{1}(Y^{\prime})$ be such that $\mathrm{spec}(\Delta_{Y^{\prime}})=\{\,0\leq\lambda_{1}\leq\lambda_{2}\leq\ldots\,\}.$ Let $G=G(\Gamma^{\prime},S)$ be the Schreier coset graph of $S$ acting on $\Gamma/\Gamma^{\prime}$. Then

[TABLE]

Theorem 1.15 was extended to Galois covers of non-elementary convex co-compact hyperbolic surfaces by Bourgain, Gamburd and Sarnak in [BGS11, Thm. 1.2] where the left hand side of (1.2) is replaced by the gap between $\delta(1-\delta)$ and the next eigenvalue of the $L^{2}$-Laplacian. This extends to non-Galois covers and therefore applies in the setting of this paper as follows.

Let us assume that $X_{n}=\Gamma_{n}\backslash\mathbb{H}$ is connected, for simplicity, although the argument can be adapted to the general case. For fixed $S\subset\Gamma$, the Schreier coset graphs $G_{n}$ of $S$ acting on $\Gamma/\Gamma_{n}\cong[n]$ are precisely the random regular graphs of the permutation model, and a.a.s. these have a uniform spectral gap by [BS87, Fri08]. Hence by the extension of [BGS11, Thm. 1.2] the $X_{n}$ have a uniform spectral gap between $\delta(1-\delta)$ and the next $L^{2}$-eigenvalue. Importantly, in all versions of Theorem 1.15, the constant $c$ depends on $Y$ in a complicated way. Because of this, it is unlikely such an argument would lead to e.g. Theorem 1.1. However, this argument does lead to Corollary 1.12 when $\delta>\frac{1}{2}$ (cf. Remark 1.13).

It is also worth mentioning that a variant of Theorem 1.15 has also been developed for resonances in [BGS11, OW16, MOW17], for specific congruence coverings of $Y=\Gamma\backslash\mathbb{H}$ where $\Gamma$ is an infinite index subgroup of $\mathrm{SL}_{2}(\mathbf{Z})$. Besides only dealing with Galois covers, the key reason that these methods cannot prove Corollary 1.12 when $\delta\leq\frac{1}{2}$ is the following. The state of the art method [MOW17, Appendix] of dealing with low frequency resonances (a la Theorem 1.8) involves bounds on the dimensions of non-trivial irreducible representations of finite groups $\mathcal{G}$ that are polynomial in $|\mathcal{G}|$. The relevant groups in our setting are $S_{n}$, and the issue is that $S_{n}$ has non-trivial irreducible representations of dimensions that are sub-logarithmic in $|S_{n}|=n!$.

Finally we point out that the methods of [BGS11, OW16, MOW17] are not well adapted to efficiently tracking constants and hence likely not suitable for producing explicit resonance free regions as in Theorem 1.8.

1.2. Overview of proofs and paper organization

All the proofs of the paper rely on a Schottky encoding of the action of $\Gamma$ on $\mathbf{R}$ that is presented in $\S$2.1. To control resonances (and eigenvalues) we rely on the connection between resonances and zeros of the Selberg zeta function due to Patterson and Perry [PP01]. This connection is explained in $\S$2.2. We then pass to dynamical considerations by the relationship between Selberg zeta functions and dynamical zeta functions explained in $\S\ref{subsec:Zeta-functions}.$ The relevant dynamical zeta functions are Fredholm determinants of certain transfer operators on vector valued functions, twisted by (random) unitary representations $\rho_{n}^{0}$ of $\Gamma$. These are introduced in $\S$2.2. The relevance of these representations is that the zeros of the $\rho_{n}^{0}$-twisted Selberg zeta function of $X$ correspond to new resonances of $X_{n}$ (see $\S$4.1). These are precisely the objects we wish to control.

**Theorems 1.1 and 1.8. **Since Theorem 1.8 implies Theorem 1.1 it suffices to discuss the former.

So far we have not been precise about the transfer operators we use. To prove Theorem 1.8 we do not use the ‘standard’ twisted transfer operators used for example in [BGS11, OW16, MOW17], but rather, we base our twisted operators on the refined transfer operators introduced by Bourgain and Dyatlov in [BD17]. The operators are denoted by $\mathcal{L}_{\tau,s,\rho_{n}^{0}}$ and defined precisely in $\S$2.2. The parameter $s$ is a frequency parameter, and the parameter $\tau$ is a ‘discretization parameter’ that is taken to be $n^{-\frac{2}{\delta}}$. If we do not use this operator in the definition of the dynamical zeta function, but rather, an iterate of the standard one, without the built in parameter $\tau$, then one can still follow the strategy of this paper to obtain resonance-free regions. However, these will depend on subtle features of the graph of the pressure functional $P(\sigma)$ defined in $\S$2.1. It is the use of refined transfer operators that allows us to improve on this, and is a key idea in the paper. The functional spaces we use are Bergman spaces, and this gives us crucial access to trace techniques.

To control zeros of the dynamical zeta function in a rectangle, we use Jensen’s formula with a circle enclosing the rectangle (cf. Figure 6.1). The strategy is to prove that the expected number of zeros in the region decays as a polynomial in $n$, so by Markov’s inequality, a.a.s. there are none. There are two terms in Jensen’s formula we need to control. The first is $\log|\det(1-\mathcal{L}_{\tau,s,\rho_{n}^{0}})|$ when $s$ is the center of the circle. As shown in Proposition 4.8, this term decays provided the center of the circle is a sufficiently large real number, which can be arranged. The second term in Jensen’s formula is the integral over $s$ in the circle of $\log|\det(1-\mathcal{L}_{\tau,s,\rho_{n}^{0}}^{2})$|. A convenient property of Jensen’s formula is that it is an integral formula, and we can take expectations inside the integral. Using Weyl’s inequality, and taking expectations, we reduce to bounding the expectation $\mathbb{E}_{n}\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$ of the squared Hilbert-Schmidt norm of $\mathcal{L}_{\tau,s,\rho_{n}^{0}}$ for $s$ on the circle. We need to prove these all decay uniformly and polynomially in $n$. This estimate is at the core of the proof, is stated precisely in Proposition 5.1, and its proof takes up $\S\ref{sec:The-expectation-of}$.

We now discuss the proof of Proposition 5.1. The first step is a formula for $\mathbb{E}_{n}\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$. This uses a deterministic expression for $\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$ involving a Bergman kernel and given in Lemma 4.7. The formula for $\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$ is a complex weighted sum of random variables $\mathrm{Tr}[\rho_{n}^{0}(\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1})]$, where $\gamma_{\mathbf{a}^{\prime}}$ and $\gamma_{\mathbf{b}^{\prime}}$ are elements of $\Gamma$. By linearity of expectations we obtain an expression for $\mathbb{E}_{n}\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$ as a weighted sum of expectations

[TABLE]

By passing to a majorant, in Lemma 5.4 we reduce our task to estimating a sum of the form

[TABLE]

where $\overline{Z}(\tau)$ is a set of words in the generators of $\Gamma$, and $\mathbf{a}^{\prime}$ is $\mathbf{a}$ with the last letter removed.

The strategy is to insert good bounds for (1.3) into (1.4) to obtain the decay we want. This is analogous to the *trace method *used to bound the spectral gap of a random graph, where $\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$ would be replaced by the trace of a power of the adjacency matrix. Indeed, the bounds we use for (1.3) go back to the paper of Broder and Shamir [BS87] who used the trace method to show that the second largest eigenvalue of a $2k$-regular random graph in the permutation model is a.a.s. $\leq 3k^{\frac{3}{4}}$. So the appearance of $\frac{3}{4}$ in Theorems 1.1 and 1.8 is similar to (ibid.).

In *(ibid.) *Broder and Shamir proved, roughly speaking, that $\mathbb{E}_{n}[\mathrm{Tr}(\rho_{n}^{0}(\gamma))]$ has a trivial bound if $\gamma$ is the identity, a better bound if $\gamma$ is a proper power of another element in $\Gamma$, and an even better bound if $\gamma$ does not fall in one of the previous two cases. We need a two sided estimate for (1.3) that can be deduced from more recent work of Puder [Pud15] and is stated in Theorem 5.2. According to the three cases above, we partition the range of summation in (1.4) into three different sets.

The hardest of these to deal with in (1.4) is the set $\mathrm{PowerPairs}(\tau)$ that consists of $\mathbf{a},\mathbf{b}\in\overline{Z}(\tau)$ such that $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is a proper power in $\Gamma$. We need to show that the contribution of this set to (1.4) has polynomial decay. We give a precise bound on $|\mathrm{PowerPairs}(\tau)|$ in Proposition 5.6; this proposition is at the core of the paper so we now explain the ideas of its proof.

Throughout the paper we work with real quantities $\Upsilon_{\mathbf{a}}$, where $\mathbf{a}$ is a word in the generators of $\Gamma$. These are defined in $\S$3. Roughly speaking, $\Upsilon_{\mathbf{a}}$ measures the size of the derivative of the associated group element $\gamma_{\mathbf{a}^{\prime}}$, and the set $\bar{Z}(\tau)$ is the set of words $\mathbf{a}$ such that $\Upsilon_{\mathbf{a}}\approx\tau$. This means that estimating $|\mathrm{PowerPairs}(\tau)|$ is roughly the same as estimating the sum

[TABLE]

the choice of the exponent $\frac{\delta}{2}$ optimizes the result we can get from this method. The key combinatorial observation we use to estimate (1.5) is that if $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is a proper power, after performing an absolutely bounded finite number of the following operations

• •

cutting the sequences $\mathbf{a}^{\prime}$ and $\mathbf{b}^{\prime}$,

• •

possibly replacing some cut sequence with its ‘mirror’,

• •

and regluing

one can form a long identical pair of sequences. This idea is performed rigorously in $\S$5.4. The result of these operations on the $\Upsilon$ is to introduce a bounded multiplicative constant, since $\Upsilon$ is roughly multiplicative (Lemma 3.4) and behaves well with respect to mirrors (Lemma 3.5). The result of obtaining the long identical pair of sequences is that we get bounds on (1.5) from the relationship between sums of $\Upsilon_{\mathbf{a}}$ and the pressure functional (Lemma 3.10).

**Theorem 1.10. **The proof of Theorem 1.10 is given in $\S$7. It is based on uniform Dolgopyat estimates for arbitrary unitary representations of $\Gamma$. We use the main result of Bourgain and Dyatlov [BD17] on Patterson-Sullivan measures and Fourier decay to provide a short and completely general proof of the uniform Dolgopyat estimates without having to rely on the more difficult technique from [Nau05a], which was also used in [OW16, MOW17].

1.3. Notation

If $U\subset\mathbf{C}$ we write $\overline{U}$ for the closure of $U$. We write $\mathbf{N}$ for the natural numbers and $\mathbf{N}_{0}=\mathbf{N}\cup\{0\}$.

1.4. Acknowledgments

We thank Benoît Collins and Doron Puder for helpful conversations related to this project. Both authors thank Semyon Dyatlov for discussions around this subject and the hospitality of IAS while attending the conference “Emerging Topics: Quantum Chaos and Fractal Uncertainty Principle” in Fall 2017. FN is supported by Institut Universitaire de France. We thank the anonymous referee for several comments that have improved the paper.

2. Preliminaries

In this paper we use the notational system for Schottky groups that is used in the papers of Dyatlov and Bourgain [BD17] and Dyatlov and Zworski [DZ17] since it is very convenient for the analysis in the sequel. We follow these papers closely in our development.

2.1. Words, encodings of Schottky groups, and pressure

Let $r\geq 2$ and $\mathcal{A}=\{1,\ldots,2r\}$. If $a\in\mathcal{A}$, then we write $\bar{a}=a+r\bmod 2r$. The setup of our paper is that we are given for each $a\in\mathcal{A}$ an open444This is a difference from the notation of [BD17] that we make the reader aware of. disc $D_{a}$ in $\mathbf{C}$ with center in $\mathbf{R}$. The closures of the discs $D_{a}$ for $a\in\mathcal{A}$ are assumed to be disjoint from one another. We let $I_{a}=D_{a}\cap\mathbf{R}$, an open interval. We write $\mathbf{D}=\cup_{a\in\mathcal{A}}D_{a}$ for the union of the discs.

We consider the usual action of $\mathrm{SL}_{2}(\mathbf{R})$ by Möbius transformations on the extended complex plane $\hat{\mathbf{C}}=\mathbf{C}\cup\{\infty\}$. We are given for each $a\in\mathcal{A}$ a matrix $\gamma_{a}\in\mathrm{SL}_{2}(\mathbf{R})$ with the properties

[TABLE]

We write $\Gamma=\langle\gamma_{a}\,:\,a\in\mathcal{A}\rangle$ for the group generated by the $\gamma_{a}$. Since the $D_{a}$ are disjoint, the Ping-Pong Lemma shows that $\Gamma$ is a free subgroup of $\mathrm{SL}_{2}(\mathbf{R})$. Any group obtained by this construction is called a *Schottky group. *It is a result of Button [But98] that if $X=\Gamma\backslash\mathbb{H}$ is a connected convex co-compact Riemann surface as in our main theorems, then $\Gamma$ is a Schottky group; we now fix $\Gamma$ and assume it arises from the above construction.

The elements of $\Gamma$ can be encoded by words in the alphabet $\mathcal{A}$ as follows. A *word *is a finite sequence

[TABLE]

such that $a_{i}\neq\overline{a_{i+1}}$ for $i=1,\ldots,n-1$. We say that $n$ is the* length* of $\mathbf{a}$ and denote this by $|\mathbf{a}|=n$. We write $\mathcal{W}$ for the collection of all words, $\mathcal{W}_{N}$ for the words of length $N$, and $\mathcal{W}_{\geq N}$ for the words of length $\geq N$. We write $\emptyset$ for the empty word and write $\mathcal{W}^{\circ}=\mathcal{W}-\{\emptyset\}$. For $\mathbf{a}=(a_{1},\ldots,a_{n}),\mathbf{b}=(b_{1},\ldots b_{m})\in\mathcal{W}$ we write

• •

${\bf a^{\prime}}=(a_{1},\ldots,a_{n-1})$ if $\mathbf{a}=(a_{1},\ldots,a_{n})$ and $n\geq 1$.

• •

$\mathbf{a}\to\mathbf{b}$ if either of $\mathbf{a}$ or $\mathbf{b}$ is empty, or else $a_{n}\neq\overline{b_{1}}$, in which case $(a_{1},\ldots,a_{n},b_{1},\ldots,b_{m})$ is in $\mathcal{W}^{\circ}$ and we write $\mathbf{a}\mathbf{b}$ for this concatenation.

• •

$\mathbf{a}\rightsquigarrow\mathbf{b}$ if $\mathbf{a},\mathbf{b}\in\mathcal{W}^{\circ}$ and $a_{n}=b_{1}$, which case $\mathbf{a}^{\prime}\mathbf{b}$ is in $\mathcal{W}^{\circ}$.

If $\mathbf{a}=(a_{1},\ldots,a_{n})\in\mathcal{W}$ then we associate to $\mathbf{a}$ the group element $\gamma_{\mathbf{a}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\gamma_{a_{1}}\ldots\gamma_{a_{n}}$; here $\gamma_{\emptyset}=\mathrm{id}.$ The map $\mathbf{a}\in\mathcal{W}\mapsto\gamma_{\mathbf{a}}\in\Gamma$ is a one-to-one encoding of $\Gamma$. We write $\overline{\mathbf{a}}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\overline{a_{n}},\ldots,\overline{a_{1}})$ and call this the mirror of $\mathbf{a}$. Note that $\gamma_{\overline{\mathbf{a}}}=\gamma_{\mathbf{a}}^{-1}$. If $\mathbf{a}=(a_{1},\ldots,a_{n})\in\mathcal{W}^{\circ}$ we let

[TABLE]

and write $|I_{\mathbf{a}}|$ for the length of the open interval $I_{\mathbf{a}}$.

The Bowen-Series map $T:\mathbf{D}\to\hat{\mathbf{C}}$ is given by

[TABLE]

The Bowen-Series map is eventually expanding [Bor16, Prop. 15.5]; this will be made explicit below so we do not give the general definition now. The limit set $\Lambda=\Lambda(\Gamma)$ of $\Gamma$, defined in the Introduction, coincides with the non-wandering set of $T$:

[TABLE]

The limit set $\Lambda$ is a compact $T$-invariant subset of $\mathbf{R}$. Given a Hölder continuous map $\varphi:\Lambda\rightarrow\mathbf{R}$, the topological pressure $P(\varphi)$ can be defined through the variational formula:

[TABLE]

where the supremum is taken over all $T$-invariant probability measures on $\Lambda$, and $h_{\mu}(T)$ stands for the measure-theoretic entropy. A celebrated result of Bowen [Bow79] says that the map

[TABLE]

is convex555Convexity follows obviously from the variational formula above. , strictly decreasing and vanishes exactly at $\sigma=\delta(\Gamma)$, the Hausdorff dimension of the limit set $\Lambda$. In addition, it is not difficult to see from the variational formula that $P(-\sigma\log|T^{\prime}|)$ tends to $-\infty$ as $\sigma\rightarrow+\infty$. For simplicity, we will use the notation $P(\sigma)$ in place of $P(-\sigma\log|T^{\prime}|)$. The pressure will play a role in some of the estimates in the sequel.

2.2. Functional spaces and transfer operators

Let $V$ be any Hilbert space. If $\Omega$ is any open subset of the complex numbers $\mathbf{C}$, we consider the Bergman space $\mathcal{H}(\Omega;V)$ that is the space of $V$-valued holomorphic functions on $\Omega$ with finite norm with respect to the given inner product

[TABLE]

Here $dm$ is Lebesgue measure on $\Omega$. If $V$ is separable, then $\mathcal{H}(\Omega;V)$ is a separable Hilbert space; in this paper $V$ will always be finite dimensional.

Of particular interest is $\mathcal{H}(\mathbf{D};V)$. This splits as an orthogonal direct sum

[TABLE]

If $\{\mathbf{e}_{k}\}_{k=1}^{\infty}$ is any orthonormal basis of $\mathcal{H}(D_{a};\mathbf{C})$, and $x_{1},x_{2}\in D_{a}$, then the sum

[TABLE]

converges and the resulting kernel is called the *Bergman kernel of $D_{a}$. *It is given by the explicit formula (cf. [Bor16, pg. 378])

[TABLE]

where $r_{a},c_{a}$ are the radius and center of $D_{a}$.

Throughout the sequel, $\rho:\Gamma\to\mathcal{U}(V)$ will be a unitary representation of the Schottky group $\Gamma$. If $Z\subset\mathcal{W}^{\circ}$ is any finite subset of words, then we define

[TABLE]

The complex power $\gamma^{\prime}_{\mathbf{a}^{\prime}}(x)^{s}$ is defined by analytic continuation using that $\gamma^{\prime}_{\mathbf{a}^{\prime}}(x)$ is positive on $I_{b}$ and never a negative real on $D_{b}$. One has $\mathcal{L}_{Z,s,\rho}:\mathcal{H}(\mathbf{D};V)\to\mathcal{H}(\mathbf{D};V)$. Certain particular choices of $Z$ are made throughout the paper. The basic type of transfer operator that is considered corresponds to the choice $Z=\mathcal{W}_{2}$. We write $\mathcal{L}_{s,\rho}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathcal{L}_{\mathcal{W}_{2},s,\rho}$. This operator can be written as

[TABLE]

In the following we follow Dyatlov and Zworski [DZ17, §2.4].

Definition 2.1**.**

A subset $Z\subset\mathcal{W}^{\circ}$ is a partition if there is $N\geq 0$ such that for all $\mathbf{a}\in\mathcal{W}$ with $|\mathbf{a}|\geq N$, there is a unique $\mathbf{b}\in Z$ that is a prefix of $\mathbf{a}$.

One particular family of partitions, introduced by Bourgain and Dyatlov [BD17], plays an important role in this paper. For any $\tau>0$ we define

[TABLE]

It is shown by Dyatlov and Zworski [DZ17, eqs (2.7), (2.15)] that this is indeed a partition. Not only is the partition $Z(\tau)$ important to us, but so too is its mirror set

[TABLE]

The reason for introducing this mirror set is to make Lemma 4.5 below work. Note that $\overline{Z}(\tau)$ may not be a partition, although this will not matter. We write $\mathcal{L}_{\tau,s,\rho}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathcal{L}_{\overline{Z}(\tau),s,\rho}$.

2.3. The representations appearing in this paper

In this paper we consider particular types of representations $\rho:\Gamma\to\mathcal{U}(V)$ as follows. We consider $n\in\mathbf{N}$ and the family of symmetric groups $S_{n}$ on $n$ letters. Let $V_{n}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\ell^{2}(\{1,\ldots,n\})$. The group $S_{n}$ has a standard representation $\mathrm{std}_{n}:S_{n}\to\mathcal{U}(V_{n})$ where $S_{n}$ acts by precomposition on $\ell^{2}$ functions $f:\{1,\ldots,n\}\to\mathbf{C}$. This representation is not irreducible, but splits as an orthogonal direct sum $\mathbf{1}\oplus V_{n}^{0}$ where $V_{n}^{0}$ is an irreducible representation of dimension $n-1$. We write $\mathrm{std}_{n}^{0}:S_{n}\to\mathcal{U}(V_{n}^{0})$ for the corresponding homomorphism of the symmetric group.

We now build a representation from a homomorphism $\phi_{n}:\Gamma\to S_{n}$. Since $\Gamma$ is free, $\phi_{n}$ is described simply by choosing the images of a generating set of $\Gamma$, which may be taken to be the $\gamma_{a}$ with $1\leq a\leq r$. We consider

[TABLE]

These depend on the choice of $\phi_{n}$. Later in the paper we will view $\phi_{n}:\Gamma\to S_{n}$ as a random homomorphism; its law is described by choosing the $\phi_{n}(\gamma_{a})$ with $1\leq a\leq r$ independently and uniformly at random with respect to the uniform measure on $S_{n}$. This gives random representations $\rho_{n}$ and $\rho_{n}^{0}$. We write $\mathbb{E}_{n}$ to refer to expectations of random variables with repect to the random representation $\rho_{n}^{0}$. For example, if $\gamma\in\Gamma$, then $\mathrm{Tr}[\rho_{n}^{0}(\gamma)]$ is a real random variable and we write $\mathbb{E}_{n}(\mathrm{Tr}[\rho_{n}^{0}(\gamma)])$ for its expectation. At other times we view $\phi_{n}$, $\rho_{n}$, $\rho_{n}^{0}$ as fixed and coupled to one another; it will be clear from the context whether we make probabilistic or deterministic statements.

2.4. Selberg zeta functions

If $X$ is any convex co-compact hyperbolic surface (not necessarily connected), then the Selberg zeta function of $X$ is defined for $\mathrm{Re}(s)>\delta$ by

[TABLE]

where $\mathcal{P}(X)$ is the collection of primitive666Primitive here means it is not an iterate of a shorter closed geodesic. closed geodesics on $X$, and $l(\gamma)$ is the length of such a geodesic. The function $Z_{X}(s)$ analytically continues to an entire function [Gui92, GLZ04]. One has the following theorem due to Patterson and Perry [PP01, Theorem 1.5] relating resonances of the Laplacian to the Selberg zeta function.

Theorem 2.2** (Patterson-Perry).**

If $X$ is any non-elementary convex co-compact hyperbolic surface, then any resonance of $X$ is a zero of $Z_{X}$. Conversely, if $s$ is a zero of $Z_{X}$ with $\mathrm{Re}(s)>0$ then $s$ is a resonance of $X$. In all cases, the order of the zero of $Z_{X}$ is equal to the multiplicity of the corresponding resonance.

We will also have a use for twisted Selberg zeta functions. If $\rho:\Gamma\to\mathcal{U}(V)$ is any finite dimensional unitary representation of $\Gamma$ then we let

[TABLE]

This converges to a holomorphic function in $\mathrm{Re}(s)>\delta$ and extends to an entire function by results in [FP17].

3. Estimates for derivatives

The following section contains certain technical but either easy or well-known estimates for derivatives of $\Gamma$ that will be used in the sequel. The fundamental estimates for derivatives of elements of $\Gamma$ are the following:

Lemma 3.1**.**

**Uniform contraction: **

There are $C=C(\Gamma)>0$ and $0<\bar{\theta}<\theta<1$ such that for all $\mathbf{a}\in\mathcal{W}$, $b\in\mathcal{A}$ with $\mathbf{a}\to b$, and $x\in D_{b}$,

[TABLE]

**Bounded distortion I: **

There is $K=K(\Gamma)>0$ such that for all $b\in\mathcal{A}$, $\mathbf{a}\in\mathcal{W}$ such that $\mathbf{a}\to b$ and all $x_{1},x_{2}\in D_{b}$,

[TABLE]

**Bounded distortion II: **

There is a constant $c=c(\Gamma)>0$ such that for $\mathbf{a}\in\mathcal{W}$, $b_{1},b_{2}\in\mathcal{A}$ with $\mathbf{a}\to b_{1},b_{2}$ and $x_{1}\in D_{b_{1}}$, $x_{2}\in D_{b_{2}}$,

[TABLE]

Proof.

The first two properties can be found in [Nau14, §2]. The last part is trivial if $\mathbf{a}=\emptyset$. Otherwise, if $|\mathbf{a}|\geq 1$ we can write $\mathbf{a}=\mathbf{a}^{\prime}a$ with $\mathbf{a}^{\prime}\in\mathcal{W}$ and $\mathbf{a}^{\prime}\to a\to b_{1},b_{2}$. Then for $x_{i}\in D_{b_{i}}$ we have

[TABLE]

We have $\frac{|\gamma_{a}^{\prime}(x_{1})|}{|\gamma_{a}^{\prime}(x_{2})|}\leq C$ by (3.1) and since now $\gamma_{a}(x_{1})$ and $\gamma_{a}(x_{2})$ are in $D_{a}$, (3.2) gives

[TABLE]

The equation (3.3) now follows. ∎

In the rest of the paper, for any $\mathbf{a}\in\mathcal{W}^{\circ}$, we define

[TABLE]

We set $\Upsilon_{\emptyset}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}1$. For $\mathbf{a}\in\mathcal{W}^{\circ}$, we have

[TABLE]

since $I_{\mathbf{a}}\subset I_{\mathbf{a}^{\prime}}$. Therefore there is $c=c(\Gamma)>0$ such that for any $\mathbf{a}\in\mathcal{W}$

[TABLE]

We next recall some useful results of Bourgain-Dyatlov from [BD17, §2].

Lemma 3.2**.**

There is a constant $K_{0}=K_{0}(\Gamma)>1$ such that for any $\mathbf{a}=(a_{1},\ldots,a_{n})\in\mathcal{W}^{\circ}$ and $x\in D_{a_{n}}$

[TABLE]

Proof.

For $x\in I_{a_{n}}$ this is [BD17, Lemma 2.5, (20)]. The more general result here follows by combining [BD17, Lemma 2.5] with the bounded distortion estimate (3.3). ∎

The following lemma is [BD17, Lemma 2.10, (30)].

Lemma 3.3**.**

There is a constant $K_{1}=K_{1}(\Gamma)>1$ such that for $\tau\in(0,1)$, for any $\mathbf{a}\in\overline{Z}(\tau)$ we have

[TABLE]

The next lemma says that $\Upsilon$ is coarsely multiplicative.

Lemma 3.4**.**

There is a constant $K_{2}=K_{2}(\Gamma)>1$ such that for all $\mathbf{a},\mathbf{b}\in\mathcal{W}^{\circ}$ with $\mathbf{a}\rightsquigarrow\mathbf{b}$

[TABLE]

and for $\mathbf{a},\mathbf{b}\in\mathcal{W}$ with $\mathbf{a}\to\mathbf{b}$

[TABLE]

Proof.

The first set of inequalities is [BD17, Lemma 2.7]. If either $\mathbf{a}$ or $\mathbf{b}$ is $\emptyset$, then (3.6) is trivially true with $K_{2}=1$. So assume $\mathbf{a},\mathbf{b}\in\mathcal{W}^{\circ}$. Then (3.6) follows by combining [BD17, Lemmas 2.6 and 2.7]. ∎

We also have the following ‘mirror’ estimate for $\Upsilon$.

Lemma 3.5** (Mirror estimate, [BD17, Lemma 2.8]).**

There is a constant $K_{3}=K_{3}(\Gamma)>1$ such that for any $\mathbf{a}\in\mathcal{W}$

[TABLE]

We now state some lemmas about the set $\overline{Z}(\tau)$.

Lemma 3.6**.**

There is a constant $C_{1}=C_{1}(\Gamma)>1$ such that for $\mathbf{a}=(a_{1},\ldots,a_{n})\in\overline{Z}(\tau)$, for any $x\in D_{a_{n}}$ we have

[TABLE]

Proof.

This follows by combining Lemmas 3.2 and 3.3. ∎

Given Lemma 3.6, we can make the following estimate on the word lengths of elements $\mathbf{a}\in\overline{Z}(\tau).$

Lemma 3.7**.**

There are constants $D=D(\Gamma)>1$ and $\kappa=\kappa(\Gamma)>0$ such that if $\mathbf{a}\in\overline{Z}(\tau)$, then

[TABLE]

Proof.

Write $\mathbf{a}=(a_{1},\ldots,a_{n})$. Pick $x\in D_{a_{n}}$. By Lemma 3.6 we have

[TABLE]

and combining this with (3.1) gives

[TABLE]

Since $|\mathbf{a}|=|\mathbf{a}^{\prime}|+1$, this gives the result after taking logarithms and rearranging. ∎

We now note

Lemma 3.8**.**

There is $0<\tau_{0}<1$ such that for $\tau<\tau_{0}$, $\overline{Z}(\tau)\subset\mathcal{W}_{\geq 2}$.

Proof.

This is a direct consequence of Lemma 3.7. ∎

Throughout the sequel, $\tau_{0}$ will always be the parameter given by Lemma 3.8. It will also be useful to know roughly how many elements there are in $\overline{Z}(\tau)$. This is given by [BD17, Lemma 2.13] (noting that $|\overline{Z}(\tau)|=|Z(\tau)|$).

Lemma 3.9**.**

There is $C_{2}=C_{2}(\Gamma)>1$ such that for $\tau\in(0,1]$

[TABLE]

To conclude this section, we record that certain sums of derivatives are related to the pressure functional.

Lemma 3.10**.**

For all $\sigma_{1},Q\in\mathbf{R}$ such that $0\leq\sigma_{1}<Q$ there is a constant $C=C(\sigma_{1},Q)>0$ such that for all $N\in\mathbf{N}_{0}$ and $\sigma\in[\sigma_{1},Q]$ we have

[TABLE]

and

[TABLE]

Proof.

The estimate (3.7) is a standard estimate that appears in [Nau14, Lemma 3.1]. The estimate (3.8) follows by combining (3.7) with Lemma 3.2 and increasing $C$. ∎

4. Transfer operators and zeta functions

4.1. Zeta functions

Lemma 4.1**.**

For any $Z\subset\mathcal{W}_{\geq 2}$, and any finite dimensional unitary representation $\rho$ of $\Gamma$, the operator $\mathcal{L}_{Z,s,\rho}$ is trace class on $\mathcal{H}(\mathbf{D};V)$.

Proof.

The proof is an easy adaptation of [Bor16, Lemma 15.7]. The condition $Z\subset\mathcal{W}_{\geq 2}$ rules out $\mathcal{L}_{Z,s,\rho}$ having any summand that acts as the identity on some $D_{a}$. ∎

Corollary 4.2**.**

Let $(\rho,V)$ be any finite dimensional unitary representation of $\Gamma$.

1. (1)

The operator $\mathcal{L}_{s,\rho}$ is trace class on $\mathcal{H}(\mathbf{D};V)$. 2. (2)

For $\tau<\tau_{0}$, the operator $\mathcal{L}_{\tau,s,\rho}$ is trace class on $\mathcal{H}(\mathbf{D};V)$.

Given Corollary 4.2 we can define zeta functions

[TABLE]

The determinants that appear here are Fredholm determinants. The reason that we have used $\mathcal{L}_{\tau,s,\rho}^{2}$ in the definition of $\zeta_{\tau,\rho}(s)$ is that it will later allow us to estimate $\log|\zeta_{\tau,\rho}(s)|$ in terms of the Hilbert-Schmidt norm of $\mathcal{L}_{\tau,s,\rho}$ rather than the trace norm (cf. (6.3)). On the other hand, we do not square $\mathcal{L}_{s,\rho}$ in the definition of $\zeta_{\rho}(s)$ so that we can access known results about $\zeta_{\rho}(s)$.

By the general theory of Fredholm determinants we have

Lemma 4.3**.**

Let $(\rho,V)$ be any finite dimensional unitary representation of $\Gamma$.

1. (1)

The function $\zeta_{\rho}(s)$ is an entire function of $s\in\mathbf{C}$ and

[TABLE] 2. (2)

If $\tau<\tau_{0}$ then $\zeta_{\tau,\rho}(s)$ is an entire function of $s\in\mathbf{C}$ and

[TABLE]

The relevance of the zeta functions $\zeta_{\rho}(s)$ are the following:

Proposition 4.4**.**

Let $\phi_{n}:\Gamma\to S_{n}$ be a fixed homomorphism, and $(\rho_{n},V_{n})$ the unitary representation corresponding to $\phi_{n}$ via (2.2). Let $X_{n}$ be the $n$-cover of $X$ corresponding to $\phi_{n}$.

1. (1)

We have $\zeta_{\rho_{n}}(s)=Z_{X,\rho_{n}}(s)=Z_{X_{n}}(s)$. 2. (2)

We have $Z_{X_{n}}(s)=Z_{X}(s)\zeta_{\rho_{n}^{0}}(s).$

Proof.

*Proof of Part 1. *A special case of a result of Jakobson, Naud, and Soares [JNS19, Prop. 2.2] for arbitrary finite-dimensional unitary representations gives

[TABLE]

where both sides are entire functions of $s$.

If $X_{n}$ is connected, then $X_{n}=\Gamma_{n}\backslash\mathbb{H}$ for some $\Gamma_{n}\leq\Gamma$ and $\rho_{n}=\mathrm{Ind}_{\Gamma_{n}}^{\Gamma}1$, the induction of the trivial representation from $\Gamma_{n}$ to $\Gamma$. In this case the Venkov-Zograf type induction formula proved by Fedosova and Pohl in [FP17, Thm. 6.1(ii)] (cf. [VZ82]) gives

[TABLE]

If $X_{n}$ is not connected, let $X_{n}^{(1)},\ldots,X_{n}^{(m)}$ denote its connected components, and let $X_{n}^{(j)}=\Gamma_{n}^{j}\backslash\mathbb{H}$ with $\Gamma_{n}^{j}\leq\Gamma$. If we let $\rho_{n}^{j}=\mathrm{Ind}_{\Gamma_{n}^{j}}^{\Gamma}1$ then we have $\rho_{n}=\bigoplus_{j=1}^{m}\rho_{n}^{j}$. Then

[TABLE]

where the first equality is by definition of the Selberg zeta functions, the second equality uses the induction formula [FP17, Thm. 6.1(ii)] and the last inequality uses the factorization formula [FP17, Thm. 6.1(i)]. Thus we have proved $\zeta_{\rho_{n}}(s)=Z_{X,\rho_{n}}(s)=Z_{X_{n}}(s)$. This proves Part 1.

*Proof of Part 2. *Using [JNS19, Prop. 2.2] again gives

[TABLE]

Since $\rho_{n}=\mathbf{1}\oplus\rho_{n}^{0}$, we have

[TABLE]

where the first equality used Part 1 of the lemma, the second used the factorization formula [FP17, Thm. 6.1(i)], and the third used (4.1). This proves Part 2. ∎

The following lemma adapts (a special case of) [DZ17, Lemma 2.4] to our vector-valued setting. The proof is essentially the same.

Lemma 4.5**.**

*For all sufficiently small $\tau>0$, if $u\in\mathcal{H}(\mathbf{D};V)$ is such that $\mathcal{L}_{s,\rho}u=u$, then

$\mathcal{L}_{\tau,s,\rho}u=u$.*

Corollary 4.6**.**

For all sufficiently small $\tau>0$, if $Z_{X_{n}}(s)=0$ and $Z_{X}(s)\neq 0$, then $\zeta_{\tau,\rho_{n}^{0}}(s)=0$.

Proof.

If $Z_{X_{n}}(s)=0$, $Z_{X}(s)\neq 0$, then by Proposition 4.4, Part 2, $\zeta_{\rho_{n}^{0}}(s)=0.$ Then by Lemma 4.3, Part 1, there is $u\in\mathcal{H}(\mathbf{D};V_{n}^{0})$ such that $\mathcal{L}_{s,\rho_{n}^{0}}u=u$. By Lemma 4.5, this implies that $\mathcal{L}_{\tau,s,\rho_{n}^{0}}u=u$, and hence $\mathcal{L}_{\tau,s,\rho_{n}^{0}}^{2}u=u$. Then by Lemma 4.3, Part 2, $\zeta_{\tau,\rho_{n}^{0}}(s)=0$. ∎

4.2. The Hilbert-Schmidt norm of the transfer operator

Corollary 4.6 reduces controlling zeros of the Selberg zeta function of $X_{n}$ that do not come from $X$ to controlling zeros of $\zeta_{\tau,\rho_{n}^{0}}(s)$. To do this, we will use Jensen’s formula, but before doing so, we collect some estimates. The first will be a pointwise lower bound on $|\zeta_{\tau,\rho}(s)|$ when $s$ is a sufficiently large real number (cf. $\S$4.3). The other will be an estimate for the expectation of the squared Hilbert-Schmidt norm $\|\mathcal{L}_{\tau,s,\rho}\|_{\mathrm{H.S.}}^{2}$ for $\rho=\rho_{n}^{0}$. One input to the latter result is a deterministic (non-random) expression for $\|\mathcal{L}_{\tau,s,\rho}\|_{\mathrm{H.S.}}^{2}$ that we give now.

Lemma 4.7**.**

Let $(\rho,V)$ be any finite dimensional unitary representation of $\Gamma.$ We have for any $s\in\mathbf{C}$ and $\tau\leq\tau_{0}$

[TABLE]

Here and henceforth we write $a\rightsquigarrow\mathbf{a}_{1},\mathbf{a}_{2}\rightsquigarrow b$ to mean that both $a\rightsquigarrow\mathbf{a}_{1}\rightsquigarrow b$ and $a\rightsquigarrow\mathbf{a}_{2}\rightsquigarrow b$.

Proof.

This is similar to arguments given by Jakobson and Naud in [JN16, pgs. 466-467]. For $a\in\mathcal{A}$, let $\{\mathbf{e}_{k}^{a}\}_{k=1}^{\infty}$ be an orthonormal basis for $\mathcal{H}(D_{a};\mathbf{C})$ and let $\{v_{j}\}_{j=1}^{\dim V}$ be an orthonormal basis for $V$. Then $\{\mathbf{e}_{k}^{a}\otimes v_{j}\>:\>a\in\mathcal{A},k\in\mathbf{N},1\leq j\leq\dim V\,\}$ is an orthonormal basis for $\mathcal{H}(\mathbf{D};V)$. We have

[TABLE]

The final application of Fubini’s theorem is justified since we assume $\tau\leq\tau_{0}$, so $\overline{Z}(\tau)\subset\mathcal{W}_{\geq 2}$, and each $\gamma_{\mathbf{a}^{\prime}_{1}},\gamma_{\mathbf{a}^{\prime}_{2}}$ maps $D_{b}$ into a compact subset of $D_{a}$, coupled with the fact that the convergence of $\sum_{k=1}^{\infty}\mathbf{e}_{k}^{\mathbf{a}}(x_{1})\overline{\mathbf{e}_{k}^{\mathbf{a}}(x_{2})}$ to $B_{D_{a}}(x_{1},x_{2})$ is uniform on compact subsets of $D_{a}$ (see, for example, [Bor16, Proof of Thm. 15.7]). ∎

4.3. A pointwise estimate for the

modulus of a zeta function

Proposition 4.8** (Pointwise bound for $|\zeta_{\tau,\rho}(s)|$).**

There is $\tau_{1}\leq\tau_{0}$ and $B\in\mathbf{R}$ with $B>2\delta$ such that if $\tau\leq\tau_{1}$, if $s\in[B,\infty)$, and $(\rho,V)$ is any finite dimensional unitary representation of $\Gamma$, we have

[TABLE]

Remark 4.9*.*

A crucial restriction in Proposition 4.8 is $\mathrm{Re}(s)>2\delta$ that results from the presence of $\mathcal{L}_{\tau,s,\rho}^{2}$ in the definition of $\zeta_{\tau,\rho}$.

Proof of Proposition 4.8.

We can write

[TABLE]

whenever the series inside the exponential is absolutely convergent. We have if $x\in D_{b}$

[TABLE]

Carefully applying the Lefschetz fixed point formula [Bor16, Lemma 15.9] now gives

[TABLE]

where $x_{\mathbf{a}_{1}^{\prime}\mathbf{a}_{2}^{\prime}\ldots\mathbf{a}^{\prime}_{2k}}\in\mathbf{R}$ is the unique attracting fixed point of $\gamma{}_{\mathbf{a}_{1}^{\prime}\mathbf{a}_{2}^{\prime}\ldots\mathbf{a}^{\prime}_{2k}}$. Let $b$ denote the last letter of $\mathbf{a}_{2k}$.

By using Lemmas 3.2 and 3.4 ($2k-1$ times) we obtain

[TABLE]

Now using Lemma 3.3 we obtain

[TABLE]

for some $K>1$. We now assume

[TABLE]

so that given $\tau\leq\tau_{1}$ we have

[TABLE]

We may also use the simple estimate $\mathrm{Tr}[\rho(\gamma_{\mathbf{a}_{1}^{\prime}\mathbf{a}_{2}^{\prime}\ldots\mathbf{a}^{\prime}_{2k}}^{-1})]\leq\dim V$. Putting this together gives

[TABLE]

Hence by Lemma 3.9 we obtain

[TABLE]

Choose $B$ such that $B>\max(1,2\delta)$ and

[TABLE]

with the effect of obtaining $|\mathrm{Tr}\mathcal{L}_{\tau,s,\rho}^{2k}|\leq(\dim V)K^{4ks}\tau^{(2s-2\delta)k}=(\dim V)(K^{4}\tau^{(2-\frac{2\delta}{s})})^{sk}$ when $s\geq B$. Now decrease $\tau_{1}$, if necessary, to ensure

[TABLE]

Note that $1-\frac{2\delta}{B}>0$, so this is indeed possible. The result of our choices is that when $s\geq B\geq 1$ and $\tau\leq\tau_{1}$

[TABLE]

so

[TABLE]

∎

5. The expectation of the Hilbert-Schmidt norm of the transfer operator

5.1. Statement of the main probabilistic estimate

The main estimate we wish to prove in this Section 5 is the following.

Proposition 5.1**.**

Given $H_{1}>0$, $\sigma_{1}>\frac{3\delta}{4}$, and $Q>\sigma_{1}$ there are constants $\epsilon=\epsilon(\Gamma,H_{1},Q,\sigma_{1})>0$, and $n_{0}=n_{0}(\Gamma,H_{1},Q,\sigma_{1})>0$ such that if $\tau=n^{-\frac{2}{\delta}}$, $n\geq n_{0}$, $s=\sigma+it$ with $\sigma\in[\sigma_{1},Q]$ and $|t|\leq H_{1}$ we have

[TABLE]

5.2. The expected value of the trace of a word

The key probabilistic estimate for $\rho_{n}^{0}$ that we use in this paper is essentially due to Broder-Shamir [BS87], and in the stronger form that we use it can be deduced from the work of Puder [Pud15]. We will explain how to deduce the result below.

Theorem 5.2** (Broder-Shamir, Puder).**

Let $\gamma\in\Gamma$ have reduced word length $t$. Then for any $n>t^{2}$

[TABLE]

Here $d(q)$ is the number of divisors of $q$.

Remark 5.3*.*

Broder and Shamir [BS87] only prove upper bounds for $\mathbb{E}_{n}(\mathrm{Tr}[\rho_{n}^{0}(\gamma)])$, whereas it is crucial for us to have upper and lower bounds, since we deal with *complex *weighted sums of the random variables $\mathrm{Tr}[\rho_{n}^{0}(\gamma)]$.

Deduction of Theorem 5.2.

Let $\gamma$ be an element of the non-abelian free group $\Gamma$ with reduced word length $t$. Note that Theorem 5.2 is trivial if $\gamma=\mathrm{id},$ so we assume this is not the case. Puder proves in [Pud15, pg. 885] that for $n>t$ one has an absolutely convergent Laurent series

[TABLE]

where each $a_{S}(\gamma)\in\mathbf{Z}$. Puder associates to $\gamma$ a quantity $\pi(\gamma)\in\mathbf{N}_{0}\cup\{\infty\}$ called the primitivity777For good reasons, ‘primitivity’ in the setting of [Pud15] does not coincide with the notion of primitive closed geodesics, although they are related. However, this is not relevant to the current proof.* rank* of $\gamma$*. *For our purposes, the only thing we need to know is that $\pi(\gamma)=0$ if and only if $\gamma=\mathrm{id}$, and $\pi(\gamma)=1$ if and only if $\gamma$ is a proper power. Puder also considers a certain finite set $\mathrm{Crit}(\gamma)$ of subgroups of the free group. Again, the only thing we need to know is that if if $\gamma=\gamma_{0}^{q}$, $q\geq 2$ and $q$ maximal, then $|\mathrm{Crit}(\gamma)|=d(q)-1$ [PP15, pg. 67].

The following facts are proven by Puder in [Pud15, pp. 885-887]:

• •

We have $a_{0}(\gamma)=1$, unless $\pi(\gamma)=1$, in which case

[TABLE]

• •

If $1\leq S<\pi(\gamma)-1$ then

[TABLE]

• •

If $\pi(\gamma)\neq 1$ then

[TABLE]

• •

For any $S\geq 0$

[TABLE]

Since $\mathrm{Tr}[\rho_{n}(\gamma)]=1+\mathrm{Tr}[\rho_{n}^{0}(\gamma)]$, if $\gamma=\gamma_{0}^{q}$, $q\geq 2$ and $q$ maximal, we have from (5.1)

[TABLE]

If $\gamma$ is neither a proper power nor the identity then the estimate is similar, but there is no $d(q)-1$ term since $\pi(\gamma)\geq 2$. ∎

5.3. Majorization of the expectation of the Hilbert-Schmidt norm

Lemma 5.4**.**

Given $Q,H_{1}>0$ there is a constant $C=C(\Gamma,H_{1},Q)$ such that if $\tau\leq\tau_{0}$ and $s=\sigma+it$ with $\sigma\in(0,Q]$ and $|t|\leq H_{1}$,

[TABLE]

Proof.

Suppose we are given $H_{1}$ as in the statement of the lemma. Taking the expectation of the expression given in Lemma 4.7 gives

[TABLE]

We wish to estimate the modulus of all quantities appearing in the integral on the right hand side. Firstly the assumption that $\tau\leq\tau_{0}$ ensures $\overline{Z}(\tau)\subset\mathcal{W}_{\geq 2}$, and so each $\gamma_{\mathbf{a}^{\prime}_{1}},\gamma_{\mathbf{a}^{\prime}_{2}}$ maps $D_{b}$ into a compact subset of $D_{a}$. It then follows from the explicit expression for the Bergman kernel in (2.1) that there is $K=K(\Gamma)>0$ such that

[TABLE]

for all $a,x,\mathbf{a}_{1},\mathbf{a}_{2}$ as in (5.3).

By definition, if $s=\sigma+it$,

[TABLE]

where $\arg$ is the principal value of the argument, $\arg:\mathbf{C}-\mathbf{R}_{\leq 0}\to(-\pi,\pi)$. Hence

[TABLE]

Therefore by Lemma 3.6 for some $c=c(H_{1},Q)>0$ we have for $|t|\leq H_{1}$

[TABLE]

for all $\mathbf{a}^{\prime}_{1}$, $x$ in (5.3), and the same for $\mathbf{a}_{2}$ in place of $\mathbf{a}_{1}$. Hence applying the triangle inequality to (5.3) and using (5.4) and (5.5), together with the fact that the $D_{b}$ have finite Lebesgue measure gives

[TABLE]

for some $C=C(\Gamma,H_{1},Q)$ whenever $|t|\leq H_{1}$ and $\tau\leq\tau_{0}$. ∎

The next step is to input the estimates of Theorem 5.2 into the estimate of Lemma 5.4. To organize the result we introduce, for each $q\in\mathbf{Z}_{\geq 2}$, the set

[TABLE]

and

[TABLE]

Notice that in the above, $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}\neq\mathrm{id}$. We will show

Lemma 5.5**.**

Given $Q,H_{1},\alpha,\epsilon>0$, there are constants $C=C(\Gamma,H_{1},Q)>0$ and $n_{0}=n_{0}(\Gamma,\epsilon,\alpha)$ such that if $\tau=n^{-\alpha}$ and $s=\sigma+it$ with $\sigma\in(0,Q]$, $|t|\leq H_{1}$, and $n\geq n_{0}$, we have

[TABLE]

Proof.

We will input Theorem 5.2 into Lemma 5.4. For this to be valid we need to control the word lengths of elements of $\overline{Z}(\tau)$. By Lemma 3.7, all $\mathbf{a}\in\overline{Z}(\tau)$ have $|\mathbf{a}|\leq c\log\tau^{-1}+\kappa$, so if $\tau=n^{-\alpha}$ with $\alpha>0$,

[TABLE]

for $n$ sufficiently large, say $n\geq n_{0}.$ In this case, if $\mathbf{a},\mathbf{b}\in\overline{Z}(\tau)$ the reduced word length of $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is

[TABLE]

so we may apply Theorem 5.2 to $\mathbb{E}_{n}[\mathrm{Tr}(\rho_{n}^{0}(\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}))]$. Moreover, if $t$ is the reduced word length of $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$, we have $t\leq 2c\alpha\log n+\kappa$ so for any $\epsilon>0$, we have

[TABLE]

when $n\geq n_{0}$, after increasing $n_{0}$ if necessary. Finally, in the case $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is a $q^{th}$ power in the free group $\Gamma$, with $q\geq 2$ we must have $q\leq t$ and so $d(q)\leq t\leq 2c\alpha\log n+\kappa\leq n^{\epsilon}$ for any $\epsilon>0$ and $n\geq n_{0}(\epsilon)$ (here we increase $n_{0}$ again if necessary).

With these estimates in hand, we partition the range of the sum of the right hand of (5.2) according to the following three cases:

• •

$\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is the identity; if this is the case then $|\mathbb{E}_{n}[\mathrm{Tr}(\rho_{n}^{0}(\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}))]|=n-1\leq n$. We observe that $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}=\mathrm{id}$ implies $\gamma_{\mathbf{a}^{\prime}}=\gamma_{\mathbf{b}^{\prime}}$, but since the map $\mathbf{a}\to\gamma_{\mathbf{a}}$ is one-to-one, this forces $\mathbf{a}^{\prime}=\mathbf{b}^{\prime}$. Therefore the number of pairs $(\mathbf{a},\mathbf{b})$ of this type is $\leq|\mathcal{A}||\bar{Z}(\tau)|\leq|\mathcal{A}|C_{2}\tau^{-\delta}$ by Lemma 3.9. So in total, these pairs contribute at most

[TABLE]

to the bound for $\mathbb{E}_{n}\|\mathcal{L}_{\tau,s,\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}$ given in (5.2).

• •

$\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is a $q$th power with $q$ maximal, $q\geq 2$. In this case, Theorem 5.2 gives

[TABLE]

for $n\geq n_{0}$. The total number of these pairs (for all possible $q$) is $|\mathrm{PowerPairs}(\tau)|$ so in total, these pairs contribute at most

[TABLE]

to (5.2).

• •

If $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is not the identity and not a proper power, then Theorem 5.2 gives

[TABLE]

We overestimate how many pairs of this kind there are by counting all pairs, of which there are $|\bar{Z}(\tau)|^{2}\leq C_{2}^{2}\tau^{-2\delta}$ by Lemma 3.9. So in total, these pairs contribute at most

[TABLE]

to (5.2).

Summing up the bounds (5.6), (5.7), and (5.8) gives the result. ∎

In the next section, we will estimate $|\mathrm{PowerPairs}(\tau)|$.

5.4. Estimating the size of $\mathrm{PowerPairs}$($\tau$)

Our goal is now to prove the following proposition controlling the size of $\mathrm{PowerPairs}(\tau)$.

Proposition 5.6**.**

For any $\epsilon>0$, there is $\tau_{2}=\tau_{2}(\Gamma,\epsilon)$ such that for $\tau\leq\tau_{2}$

[TABLE]

In the remainder of this $\S$5.4 we prove Proposition 5.6.

We decompose $\mathrm{PowerPairs}(\tau)$ as follows. We introduce integer parameters $L,R\geq 0$, $M_{1},M_{2}\geq 0$, and $q\geq 2$. For such parameters, let \mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q) be the subset of $\mathrm{PowerPairs}(\tau;q)$ consisting of those $(\mathbf{a},\mathbf{b})\in\mathrm{PowerPairs}(\tau;q)$ with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since every element of $\mathrm{PowerPairs}(\tau)$ belongs to some \mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q), we have

[TABLE]

We will estimate |\mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q)| in the following lemma.

Lemma 5.7**.**

*There are constants $D>0$ and $\kappa\in\mathbf{R}$ depending only on $\Gamma$ such that

\mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q) is empty unless*

[TABLE]

and

[TABLE]

Under the same assumptions as Proposition 5.6, and assuming (5.10) holds, there is a constant $K=K(\Gamma)>0$ such that

[TABLE]

Proof.

For the first statement of the lemma, if (\mathbf{a},\mathbf{b})\in\mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q), since $|\mathbf{a}^{\prime}|=N_{1}$, $|\mathbf{b}^{\prime}|=N_{2}$, and $\mathbf{a},\mathbf{b}\in\overline{Z}(\tau)$, Lemma 3.7 implies (5.10) must hold; therefore

\mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q) is empty if (5.10) does not hold. Moreover since $2\leq q\leq N_{1}+N_{2}$, after doubling $D$ and $\kappa$, (5.11) must hold also.

Now we prove (5.12) assuming (5.10) holds. For (\mathbf{a},\mathbf{b})\in\mathrm{PowerPairs}(\text{\tau,}L,M_{1},M_{2},R;q) we have

[TABLE]

where $|\mathbf{c}|=L$, $|\mathbf{d}|=R$, $|\mathbf{A}|=M_{1}$ and $|\mathbf{B}|=M_{2}$. Therefore

[TABLE]

Since $\gamma_{\mathbf{a}^{\prime}}\gamma_{\mathbf{b}^{\prime}}^{-1}$ is a $q$th power, with $q$ maximal, $\gamma_{\mathbf{A}}\gamma_{\mathbf{B}}^{-1}$ is also a $q$th power, with $q$ maximal, as they are conjugate in $\Gamma$. Since the first letters of $\mathbf{A}$ and $\mathbf{B}$ are not the same, and the last letters of $\mathbf{A}$ and $\mathbf{B}$ are not the same, we have $\mathbf{A}\to\overline{\mathbf{B}}\to\mathbf{A}$, in other words, the word $\mathbf{A}\overline{\mathbf{B}}$ is cyclically reduced. It now follows that there is some $\mathbf{u}\in\mathcal{W}^{\circ}$ with $|\mathbf{u}|=\frac{M_{1}+M_{2}}{q}$ and $\mathbf{u}\to\mathbf{u}$ (i.e. $\mathbf{u}$ is also cyclically reduced) such that

[TABLE]

$\mathbf{A}\overline{\mathbf{B}}$ is $q$ repeated copies of $\mathbf{u}$. Therefore

[TABLE]

with $\mathbf{v}_{1}\to\overline{\mathbf{v}_{2}}$ and

[TABLE]

[TABLE]

Our estimates will crucially rely on the observation that for fixed $L,M_{1},M_{2},R,q$, choosing $\mathbf{c},\mathbf{d},\mathbf{u}$ specifies $\mathbf{a}^{\prime}$ and $\mathbf{b}^{\prime}$ and hence specifies $\mathbf{a}$ and $\mathbf{b}$ except for their last letters.

We will use the shorthand $\mathbf{u}^{m}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\underbrace{\mathbf{u}\mathbf{u}\ldots\mathbf{u}}_{m}$ for $m\in\mathbf{N}$. From (5.13), using Lemma 3.4 three times gives

[TABLE]

and using the same estimate in addition to the mirror estimate of Lemma 3.5 gives

[TABLE]

Therefore, now using Lemma 3.4 in the opposite direction together with (5.14) we obtain

[TABLE]

To exploit these estimates, we note that by Lemma 3.3, we have

[TABLE]

where the last inequality is by (3.4). Let

[TABLE]

so we have

[TABLE]

By (5.15)

[TABLE]

By Lemma 3.10 there is some $C=C(\Gamma)>0$ such that

[TABLE]

To deal with $\sum_{\mathbf{u}\in\mathcal{W}_{(M_{1}+M_{2})/q}}\Upsilon_{\mathbf{u}^{q}}^{\frac{\delta}{2}}$, we write

[TABLE]

where $r=1$ if $q$ is odd and $r=0$ if $q$ is even. Now using Lemma 3.4 twice and the uniform bound for $\Upsilon_{\mathbf{u}}$ from (3.5) we obtain

[TABLE]

Therefore

[TABLE]

where the final inequality is by Lemma 3.10 and $C=C(\delta)$ is the constant provided there. Therefore in total, inputting our bounds (5.18) and (5.19) into (5.17) we get

[TABLE]

for $\tilde{K}=\tilde{K}(\Gamma)>0$. Hence by (5.16)

[TABLE]

for some $K=K(\Gamma)>0$. ∎

Now we can prove Proposition 5.6.

Proof of Proposition 5.6.

Combining (5.9) with Lemma 5.7 we obtain for constants $D,\kappa>0$

[TABLE]

for any $\epsilon>0$ and $\tau$ sufficiently small. ∎

5.5. Proof of Proposition 5.1

Suppose we are given parameters $Q,H_{1},\sigma_{1}$ as in Proposition 5.1. We assume $\sigma_{1}>\frac{3\delta}{4}$, $\sigma\in[\sigma_{1},Q]$, and $|t|\leq H_{1}$. Let

[TABLE]

Let $\alpha=\frac{2}{\delta}$ and let $\tau=n^{-\alpha}=n^{-\frac{2}{\delta}}$. Let $n_{0}$ be such that for $n\geq n_{0}$, $\tau\leq\tau_{2}$ where $\tau_{2}$ is the one provided by Proposition 5.6 for the current $\epsilon$, and $n_{0}$ is at least the one provided by Lemma 5.5 for the current $\epsilon$ and $\alpha$.

Combining Proposition 5.6 and Lemma 5.5 gives for $n\geq n_{0}$

[TABLE]

after possibly increasing $n_{0}$. This completes the proof of Proposition 5.1. $\square$

6. Proof of Theorems 1.1 and 1.8

As explained in the Introduction, Theorem 1.8 implies Theorem 1.1, so we will prove Theorem 1.8. A direct proof of Theorem 1.1 would use most of the same ideas and not be significantly shorter.

Let $n\in\mathbf{N}$, $\phi_{n}$ be a random homomorphism $\phi_{n}:\Gamma\to S_{n}$, and $(\rho_{n}^{0},V_{n}^{0})$ be the random representation described in $\S$2.3. Let $X_{n}$ be the random convex co-compact hyperbolic surface described in the Introduction.

Let $\sigma_{0}\in(\frac{3}{4}\delta,\delta)$ and $H$ be the number given in the assumptions of Theorem 1.8. Let $\tau_{1}$ and $B>2\delta$ be the constants provided by Proposition 4.8 and choose $b\geq B$ such that the open disc $D_{b-\frac{3}{4}\delta}(b)$ contains $\mathrm{Rect}(\sigma_{0},H)$. We let $\tau=n^{-\frac{2}{\delta}}$.

Since as $n$ varies in $\mathbf{N}$ and $\phi_{n}$ runs over all homomorphisms from $\Gamma\to S_{n}$, the countable collection of holomorphic functions $\zeta_{\tau,\rho_{n}^{0}}$ have amongst them all, a countable number of zeros in the closed disc $\overline{D_{b-\frac{3}{4}\delta}(b)}$, it is possible to find a $\sigma_{1}\in(\frac{3}{4}\delta,\sigma_{0})$ such that

• •

no $\zeta_{\tau,\rho_{n}^{0}}$ has a zero $s$ with $|s-b|=b-\sigma_{1},$ and

• •

the open disc $D_{b-\sigma_{1}}(b)$ contains the closed rectangle $\mathrm{Rect}(\sigma_{0},H)$

We pick such a $\sigma_{1}.$ Now we let

[TABLE]

We will shortly apply Proposition 5.1 with $Q=b+R$, $\sigma_{1}$ as is it is in the current context, and $H_{1}=R$. Let $\epsilon$ and $n_{0}$ be the positive constants provided by these inputs to Proposition 5.1. We pick $n_{1}\geq n_{0}$ such that for $n\geq n_{1}$, $\tau\leq\tau_{1}$. This sets up all the constants for the proof.

If for $\sigma>0$, $\sigma$ is an resonance for $X_{n}$, and is either not a resonance of $X$ or a resonance of $X$ with a lower multiplicity, then by Theorem 2.2 combined with Corollary 4.6, $\zeta_{\tau,\rho_{n}^{0}}(\sigma)=0$. *Therefore it suffices to show that a.a.s. there are no zeros of $\zeta_{\tau,\rho_{n}^{0}}$ in $\mathrm{Rect}(\sigma_{0},H)$. *

Let $\mathcal{N}(\phi_{n})$ be the number of zeros of $\zeta_{\tau,\rho_{n}^{0}}$ in $\mathrm{Rect}(\sigma_{0},H)$. Note that $\mathrm{Rect}(\sigma_{0},H)\subset\overline{D_{R^{\prime}}(b)}$. By Jensen’s formula [Bor16, Thm. A.2] applied to the translate of $\zeta_{\tau,\rho_{n}^{0}}$ by $b$ we have

[TABLE]

The star on the sum means zeros are repeated according to their multiplicity. Note that $b\geq B$ so Proposition 4.8 ensures $\zeta_{\tau,\rho_{n}^{0}}(b)\neq 0$, and the choice of $\sigma_{1}$ ensures $\zeta_{\tau,\rho_{n}^{0}}(b+Re^{i\theta})$ is never zero. These conditions were needed for Jensen’s formula. Now (6.1) implies

[TABLE]

Next we majorize $\mathcal{M}(\phi_{n})$. By Weyl’s inequality (cf. [Bor16, (A36)]) we have for any $s\in\mathbf{C}$

[TABLE]

where $\|\bullet\|_{1}$ and $\|\bullet\|_{\mathrm{H.S.}}$ stand for the trace and Hilbert-Schmidt norms, respectively. This was the reason for the square in the definition of $\zeta_{\tau,\rho_{n}^{0}}(s)$. Also, by Proposition 4.8 we have

[TABLE]

since $\delta\in(0,1)$. Using (6.3) and (6.4) gives

[TABLE]

Combining (6.2) and (6.5) and taking expectations gives

[TABLE]

By Proposition 5.1 we have $\mathbb{E}_{n}[\|\mathcal{L}_{\tau,b+Re^{i\theta},\rho_{n}^{0}}\|_{\mathrm{H.S.}}^{2}]\leq n^{-\epsilon}$ for all $\theta\in[0,2\pi].$ Hence

[TABLE]

for $n\geq n_{1}$. By Markov’s inequality, the probability that $\zeta_{\tau,\rho_{n}^{0}}$ has at least one zero in $\mathrm{Rect}(\sigma_{0},H)$ is bounded by the right hand side of (6.6); since this $\to 0$ as $n\to\infty$, a.a.s. $\zeta_{\tau,\rho_{n}^{0}}$ has no zeros in $\mathrm{Rect}(\sigma_{0},H)$. Hence by the previous arguments, a.a.s.

[TABLE]

and the multiplicities on both sides are the same. $\square$

7. Proof of Theorem 1.10 about high frequency resonances

This part is largely independent from the previous sections. Although we use the technique of induced representations to keep track of resonances in covers, we prove a spectral estimate on transfer operators twisted by any unitary representation which implies Theorem 1.10 via induced representations. We will prove the following completely general fact. Let $\rho:\Gamma\rightarrow\mathcal{U}(V)$ be a unitary representation of $\Gamma$ on a complex Hilbert888We do not assume that it is finite dimensional here. space $V$. Here $\mathcal{U}(V)$ is the set of unitary operators on $V$. Recall that $I=\cup_{j=1}^{2r}I_{j}$. Let $C^{1}(I,V)$ denote the Banach space of $V$-valued functions, $C^{1}$ on $\overline{I}$, endowed with the norm ($t\neq 0$)

[TABLE]

where as usual

[TABLE]

where $\|.\|_{V}$ is the Hilbert space norm on $V$. We recall that the action of the “basic” transfer operator $\mathcal{L}_{s,\rho}$, now on the function space $C^{1}(I,V)$, is given by

[TABLE]

We will use the notation $\mathcal{W}_{N}^{j}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left\{{\bf a}\in\mathcal{W}_{N}\ :\ {\bf a}\rightarrow j\right\}$. Given the previously defined notations and $F\in C^{1}(I,V)$, we have for all $x\in I_{j}$ and $N\in\mathbf{N}$,

[TABLE]

We mention here that we could also alternatively use the “refined” transfer operator $\mathcal{L}_{\tau,s,\rho}$ here in place of $\mathcal{L}_{s,\rho}^{N}$, but it wouldn’t change the final result, nor it would make the size of the gap explicit. We will need in this section some standard distortion estimates. Some of them (bounded distortion) have already been used in previous sections, but we recall them for the convenience of the reader.

• •

(Uniform hyperbolicity). There exists $C>0$ and $0<\overline{\theta}<\theta<1$ such that for all $N$ and all $j$ such that ${\bf a}\in\mathscr{\mathcal{W}}_{N}^{j}$, then for all $x\in I_{j}$ we have

[TABLE]

• •

(Bounded distortion). There exists $M_{1}>0$ such that for all $N,j$ and all ${\bf a}\in\mathscr{\mathcal{W}}_{N}^{j}$,

[TABLE]

• •

(Bounded distortion for the third derivatives). There exists $Q>0$ such that for all $n,j$ and all ${\bf a}\in\mathscr{\mathcal{W}}_{N}^{j}$,

[TABLE]

Notice that the “bounded distortion for the third derivatives” follows directly from differentiating two times $\log(\gamma^{\prime}_{{\bf a}})$, and using bounded distortion and uniform hyperbolicity several times, see for example [BV05, §3] for a previous occurrence of this condition in the literature. We now state the Ruelle-Perron-Frobenius Theorem, which will be used below. The statement of this theorem in the symbolic setting can be found in [PP90, Thm. 2.2]. The version we use can be obtained via the work of Liverani [Liv95] as in [Nau05a, Thm. 5.1].

Theorem 7.1**.**

Set $\mathcal{L}_{\sigma}=\mathcal{L}_{\sigma,\mathrm{Id}}$ where $\sigma$ is real and $\rho=\mathrm{Id}$ means the trivial one-dimensional representation.

1. (1)

The spectral radius of $\mathcal{L}_{\sigma}$ on $C^{1}(I,\mathbf{C})$ is $e^{P(\sigma)}$ which is a simple eigenvalue associated to a strictly positive eigenfunction $h_{\sigma}>0$ in $C^{1}(I,\mathbf{C})$. 2. (2)

The operator $\mathcal{L}_{\sigma}$ on $C^{1}(I,\mathbf{C})$ is quasi-compact with essential spectral radius smaller than $\kappa(\sigma)e^{P(\sigma)}$ for some $\kappa(\sigma)<1$. 3. (3)

There are no other eigenvalues on $|z|=e^{P(\sigma)}$. Moreover, the spectral projector $\mathbb{P}_{\sigma}$ on $\{e^{P(\sigma)}\}$ is given by

[TABLE]

where $\mu_{\sigma}$ is the unique probability measure on $\Lambda$ that satisfies $\mathcal{L}_{\sigma}^{*}(\mu_{\sigma})=e^{P(\sigma)}\mu_{\sigma}$, and the eigenfunction $h_{\sigma}$ is normalized so that

[TABLE]

We continue with a basic a priori estimate.

Lemma 7.2**.**

Fix some $\sigma_{0}<\delta$, then there exists $C_{0}>0,\ \rho<1$ such that for all $N$, all unitary representations $(\rho,V)$ and all $s=\sigma+it$ with $\sigma\geq\sigma_{0}$, we have

[TABLE]

Proof.

Differentiate the formula for $\mathcal{L}_{s,\rho}^{n}(f)$: since the representation factor is locally constant, we don’t need to differentiate it. Use the bounded distortion property plus the uniform contraction, combined with the pressure estimate in Lemma 3.10. Uniformity with respect to $(\rho,V)$ follows from triangle inequality plus the fact that for all $\gamma\in\Gamma$, we have $\|\rho(\gamma)\|_{V}=1$. ∎

The main fact of this section is the following. It is essentially a vector-valued version of a result stated in [JNS19]. This type of estimate is called a *Dolgopyat estimate *by reference to Dolgopyat’s work on Anosov flows [Dol98] where these type of bounds appeared for the first time.

Proposition 7.3**.**

There exist $\varepsilon>0$, $T_{0}>0$ and $C_{1},\beta>0$ such that for all $N=N(t)=[C_{1}\log|t|]$ with $s=\sigma+it$ satisfying $|\sigma-\delta|\leq\varepsilon$ and $|t|\geq T_{0}$, we have

[TABLE]

All the constants here are uniform with respect to $\rho,V$.

A particular case of this estimate was proved in [OW16, MOW17] for the case of congruence subgroups, where

[TABLE]

is obtained after reduction mod $p$ via the regular representation of $\mathrm{SL}_{2}(\mathbb{F}_{p})$. The proof was an adaptation of the arguments of [Nau05a]. We will present below a shorter, more direct version of this estimate which allows to prove this generalization without much effort.

Let us first briefly explain why this actually implies Theorem 1.10. We set $\rho=\mathrm{Ind}_{\widetilde{\Gamma}}^{\Gamma}$, where $\widetilde{\Gamma}$ is an arbitrary, finite index subgroup of $\Gamma$, and $\mathrm{Ind}_{\widetilde{\Gamma}}^{\Gamma}$ is the induced representation to $\Gamma$ of the trivial representation of $\widetilde{\Gamma}$. We work by contradiction. Assume that $Z_{\widetilde{\Gamma}}(s)=0$, then according to the induction formula of Venkov-Zograf [VZ82, FP17], we have for $s=\sigma+it$,

[TABLE]

for some $F_{s}\not\equiv 0\in C^{1}(I,V)$. We can definitely normalize $F_{s}$ so that $\|F_{s}\|_{(t),V}=1$. Write $N=N_{1}+N(t)$, where $N(t)$ is given by Proposition 7.3. Take $\sigma_{0}\leq\sigma\leq\delta$. Using the triangle inequality for $\|.\|_{V}$ and unitarity of $\rho$, we have (by Cauchy-Schwarz) and the pressure estimate (Lemma 3.10),

[TABLE]

We need to estimate the $C^{1}$-norm of $x\mapsto\|\mathcal{L}_{s,\rho}^{N(t)}(F_{s})\|_{V}^{2}(x)$ on $I$. Since we work with a Hilbert norm, the square of the norm is differentiable and we can compute

[TABLE]

and use the $V$-valued Lasota-Yorke estimate from Lemma 7.2 and Cauchy-Schwarz to obtain

[TABLE]

Using the Ruelle-Perron-Frobenius Theorem (Theorem 7.1), and the fact that $P(\delta)=0$, we get

[TABLE]

with $\kappa_{\delta}<1$. Assuming that $\sigma_{0}\geq\delta-\epsilon$ and $|t|\geq T_{0}$, we can apply Proposition 7.3 and set $N_{1}=N_{1}(t)=[C_{2}\log|t|]$ to get

[TABLE]

We then take $C_{2}$ large enough and fix $\sigma_{0}$ close enough to $\delta$ so that $C_{2}P(2\sigma_{0}-\delta)-\beta<0$ and we get

[TABLE]

for some $\widetilde{\beta}>0$. The same calculation can be performed to obtain similarly

[TABLE]

and we reach a contradiction for all $|t|$ large since $1=\|F_{s}\|_{(t),V}\leq C^{\prime}|t|^{-\widetilde{\beta}/2}.$ Once again, all the constants are uniform with respect to $(\rho,V)$.

The proof of the key Proposition 7.3 will rest on the following result of Bourgain-Dyatlov [BD17].

Theorem 7.4**.**

There exist constants $\beta_{1},\beta_{2}>0$ such that the following holds. Given $g\in C^{1}(I)$ and $\Phi\in C^{2}(I)$, consider the integral

[TABLE]

If we have

[TABLE]

and $\|\Phi\|_{C^{2}}\leq M$, then for all $|\xi|\geq 1$, we have

[TABLE]

where $C_{M}>0$ does not depend on $\xi,g$.

For comments on this version of the Bourgain-Dyatlov decay estimate, see [JNS19]. Let us just mention that $\mu_{\delta}$, up to a smooth density, is the Patterson-Sullivan measure, see [JNS19]. To be able to use this estimate, we will use the following fact from [JNS19], which is referred there as the “uniform-non-integrability property” (UNI), see Proposition 4.10.

Proposition 7.5**.**

(UNI) For all ${\bf a},{\bf b}\in\mathscr{\mathcal{W}}_{N}^{j}$ set

[TABLE]

There exist constants $M>0$ and $\eta_{0}>0$ such that for all $n$ and all $\epsilon=e^{-\eta N}$ with $0<\eta<\eta_{0}$, we have for all ${\bf a}\in\mathscr{\mathcal{W}}_{N}^{j}$,

[TABLE]

For a proof of that fact, see [JNS19, §4]. We are now ready to conclude this section by the proof of Proposition 7.3. Pick $f\in C^{1}(I,V)$. We set $s=\sigma+it$ and we assume that $\sigma$ is close to $\delta$. Let us write

[TABLE]

with

[TABLE]

and

[TABLE]

Notice that $g_{{\bf a},{\bf b}}^{(j)}$ is indeed a $C^{1}$ function on a neighborhood of $\Lambda(\Gamma)$. By using the bounded distortion property and Cauchy-Schwarz we have easily:

[TABLE]

Differentiating inside the inner product $\langle.,.\rangle_{V}$ and using the bounded distortion plus the uniform contraction (with Cauchy-Schwarz again) gives also

[TABLE]

Both estimates (7.1) and (7.2) can be combined to yield

[TABLE]

We also observe that $\inf_{x\in I_{j}}|\Phi_{{\bf a},{\bf b}}^{\prime}(x)|=\mathscr{D}({\bf a},{\bf b})$, and that by using the bounded distortion for the second and third derivatives we have for some uniform $C_{3}>0$,

[TABLE]

The plan is now to split $\mathcal{S}_{\sigma,N}(t)$ as

[TABLE]

with the “near-diagonal” sum

[TABLE]

and the “off-diagonal“ sum

[TABLE]

with $\epsilon>0$. We now assume that $\sigma_{0}\leq\sigma\leq\delta$ and $N=[\kappa\log|t|]$, with $\epsilon=e^{-\eta N}$with $0<\eta<\eta_{0}$. We fix $\kappa$ large enough so that $|t|\theta^{N}$stays uniformly bounded as $|t|\rightarrow\infty$, and pick $\eta>0$ small enough such that $\epsilon=e^{-\eta[\kappa\log|t|]}>|t|^{{}^{-\beta_{1}}}$ , so that we can apply Theorem 7.4. Combining estimate (7.3) with the pressure bound from Lemma 3.10, we get

[TABLE]

On the other hand we have

[TABLE]

which by using Proposition 7.5 and the pressure estimate combined with the uniform hyperbolicity (the lower bound) gives

[TABLE]

because $P(\sigma_{0})\rightarrow 0$ as $\sigma_{0}\rightarrow\delta$, we can definitely pick $\sigma_{0}<\delta$ so that for all $|t|\geq 1$, we have

[TABLE]

for some uniform $\widetilde{C}>0$ and $\widetilde{\beta}>0$. This ends the proof.