Expansion and uniform resonance free regions for convex cocompact hyperbolic surfaces
aa r X i v : . [ m a t h . SP ] J a n UNIFORM RESONANCE FREE REGIONS FOR CONVEXCOCOMPACT HYPERBOLIC SURFACES AND EXPANDERS
LOUIS SOARES
Abstract.
Let X = Γ \ H be a convex cocompact hyperbolic surface, let X n =Γ n \ H be an infinite family of finite regular covers of X , and let G n be the associatedfamily of Cayley graphs of the covering groups G n = Γ / Γ n with respect to the set ofSchottky generators of X . Motivated by the work of Brooks [12], Burger [13, 14], andBourgain–Gamburd–Sarnak [7], we conjecture that the surfaces X n have a uniform resonance-free strip if the graphs G n form a family of expanders . We prove someresults towards this conjecture. In particular, we confirm the conjecture under twoadditional assumptions: the G n ’s form a family of two-sided expanders and the non-trivial irreducible representations of the groups G n are “large” compared to thesize of G n . Combining this result with the work on expanders by Breuillard–Green–Guralnick–Tao [11], we show that for every finite simple group of Lie type G , “almostall” G -covers of X have a uniform resonance-free strip. Another consequence of ourresults is a new proof of the result of Oh–Winter [31] on uniform resonance-free stripsfor congruence covers of convex cocompact hyperbolic surfaces. Our proofs rely onrepresentation theoretic and transfer operator techniques. Introduction
Spectral gap and expanders.
The relation between spectral gaps of the Lapla-cian on compact Riemannian manifolds and expander graphs is well-known in theliterature, mainly through the work of Brooks [12] and Burger [13, 14].Expanders are highly-connected sparse graphs widely used in computer science withmany applications in pure mathematics. There is a vast literature on expanders andwe refer the reader to [20, 25, 39, 23] for a comprehensive discussion. For an undirectedgraph G with vertex set V and a subset A ⊂ V , the boundary ∂A of A is defined tobe the set of edges with one extremity in A and the other in V r A . The Cheegerisoperimetric constant of G is defined as h ( G ) = min (cid:26) | ∂A || A | : A ⊂ V, ≤ | A | ≤ | V | (cid:27) . A family of k -regular graphs G n is said to form a family of expanders if there exists c > n we have h ( G n ) ≥ c .Let X be a compact Riemannian manifold, let X n be an infinite family of finite cov-erings of X , and let S = { γ ± , . . . , γ ± m } be a fixed set of generators of the fundamentalgroup π ( X ). The fundamental group π ( X n ) of each X n can be viewed as a subgroupof π ( X ). Now we consider the family of graphs G n defined as follows: the vertices of G n are the cosets of π ( X ) /π ( X n ) (of which there are finitely many) and two verticesof G n are joined by an edge if the corresponding cosets differ by left multiplication by Mathematics Subject Classification.
Primary: 58J50, 11M36, Secondary: 37C30, 37D35,05C90.
Key words and phrases.
Resonances, hyperbolic surfaces, expanders, Selberg zeta function, transferoperators. one element in the generating set S . If we assume furthermore that the X n ’s are regular coverings of X , then we can view each π ( X n ) as a normal subgroup of π ( X ). In thiscase the quotient G n = π ( X ) /π ( X n ) forms a group and the graphs G n are the Cayleygraphs of the group G n with respect to the set p n ( S ) ⊂ G n , where p n : π ( X ) → G n isthe natural projection.We can now state the aforementioned result of Brooks [12] and Burger [13, 14]: theLaplacians acting on the X n ’s have a uniform positive spectral gap if and only if thegraphs G n form a family of expanders . Bourgain–Gamburd–Sarnak [7] extended thisresult to the case where X is a geometrically finite hyperbolic surface with cusps . Wenote that if X is either a compact Riemannian manifold or a hyperbolic manifold X with cusps, we say that a family X n of finite covers of X has a uniform spectral gap ifthere exists some constant c > n , we have λ ( X n ) ≥ λ ( X n ) + c, where 0 ≤ λ ( X n ) < λ ( X n ) ≤ · · · is the sequence of L -eigenvalues of the positive Laplacian on X n .In this paper, we focus on the case where X = Γ \ H is a convex cocompact hyperbolicsurface, that is, a non-compact hyperbolic surface with a finite number of funnel endsand no cusps (see [3]). We conjecture that a similar relation between expanders andspectral gap remains true, under a suitable reformulation of the notion of spectral gapin terms of the resonances of X .Let us briefly recall some key facts on the spectral theory of infinite-area hyperbolicsurfaces, referring the reader to Borthwick’s book [3] for more details. For the remain-der of this introduction we assume that Γ is a finitely generated, non-cofinite Fuchsiangroup and we let X = Γ \ H be the associated hyperbolic surface. The limit set Λ Γ is defined as the set of accumulation points of orbits of Γ acting on the hyperbolicplane H . It is a subset of the boundary of H at infinity, which we identify with R = R ∪ {∞} . The limit set is a Cantor-like fractal whose Hausdorff dimension wedenote by δ Γ . The value δ Γ coincides with the exponent of convergence of Poincar´eseries for Γ.The L -spectrum of the positive Laplacian ∆ X on X has been described completely byLax–Phillips [24]. The continuous spectrum of ∆ X coincides with the interval [1 / , ∞ ),which does not contain any embedded L -eigenvalues. The pure point spectrum isempty if δ Γ ≤ , and finite and starting at δ Γ (1 − δ Γ ) if δ Γ > . This description ofthe spectrum implies that the resolvent R X ( s ) = (∆ X − s (1 − s )) − : L ( X ) → L ( X )is well-defined and analytic on the half-plane { Re( s ) > } except at the finite set ofpoles corresponding to the pure point spectrum of ∆ X . The resonances of X are thepoles of the meromorphic continuation of(1) R X ( s ) : C ∞ ( X ) → C ∞ ( X )to the whole complex plane. This continuation can be deduced using the analytic Fred-holm theorem together with an adequate parametrix provided by Guillop´e–Zworski[19]. In the sequel, we denote by R X the set of resonances of X . The resolvent oper-ator (1) has a simple pole at s = δ Γ and no other poles in the half-plane Re( s ) ≥ δ Γ ,by a result of Patterson [33]. Put differently, s = δ Γ is the resonance of X with thelargest real part. By the work of Patterson–Perry [34], resonances appear as zeros of the
Selberg zetafunction , which is defined for Re( s ) ≫ Z Γ ( s ) := ∞ Y k =0 Y [ γ ] ∈ [Γ] p (cid:16) − e − ( s + k ) ℓ ( γ ) (cid:17) . Here, the inner product is taken over the set [Γ] p of conjugacy classes of primitivehyperbolic elements of Γ.By the work of Naud [26], every non-elementary, convex cocompact X has a spectralgap in the following sense: there exists some η > X in the half-plane Re( s ) > δ Γ − η , except for the resonance at s = δ Γ (see also themore recent work of Bourgain–Dyatlov [4]). Finding resonance-free regions has a longhistory and applications in many settings. In this paper we are interested in finding uniform resonance-free regions in infinite families of finite degree covers of X .To properly state the conjecture we alluded to above, we recall the result of Button [15]which says that every convex cocompact hyperbolic surface X is isometric to a quotientΓ \ H where Γ is a Schottky group . Schottky groups stand out, among other Fuchsiangroups, by their simple geometric construction (see § S = { γ , . . . , γ m } the set of Schottky generators for Γ arising from thisconstruction. Now we consider an infinite family of finite-index normal subgroups Γ n ofΓ and we let X n = Γ n \ H be the associated covers of X . Moreover, we let G n = Γ / Γ n be the (finite) covering groups and we let G n be the associated family of Cayley graphs G n = Cay( G n , S ) , that is, the vertices of G n are the elements of the group G n and two vertices x, y ∈ G n are joined by an edge if and only if there exists some s ∈ S with y = p n ( s ) x , where p n : Γ → G n is the natural projection map.With this in place we can now state our conjecture: Conjecture 1.1.
Let Γ be a non-elementary Schottky group, let Γ n be an infinitefamily of finite-index, normal subgroups of Γ , let G n = Γ / Γ n be the associated coveringgroups, and let G n = Cay( G n , S ) denote the graphs constructed above. Assume that G = ( G n ) n forms a family of expanders. Then there exists η > (depending only on G and Γ ) such that for each n , the Selberg zeta function Z Γ n ( s ) has no non-trivial zerosin the half-plane Re( s ) ≥ δ Γ − η. Here, s ∈ C is said to be a “trivial” zero of Z Γ n ( s ) if it is a zero of Z Γ ( s ) . Equivalently,the surfaces X n = Γ n \ H have a uniform resonance-free strip, that is, for each n , wehave R X n ∩ { Re( s ) ≥ δ Γ − η } = R X ∩ { Re( s ) ≥ δ Γ − η } . Remark . • A Fuchsian group Γ is said to be “non-elementary” if it is generated by morethan one element. • For every finite-index, normal subgroup Γ ′ of Γ we have Λ Γ ′ = Λ Γ . In par-ticular, for any given family Γ n satisfying the assumptions of Conjecture 1.1,we have δ Γ n = δ Γ , and the point s = δ Γ is a common simple resonance ofthe surfaces X n and therefore also a common simple zero of the Selberg zetafunction Z Γ n ( s ) . L. SOARES • By the Venkov–Zograf factorization formula in § Z Γ ( s ) is alsoa zero of Z Γ n ( s ). Consequently, the non-trivial zeros of Z Γ n ( s ) are preciselythose resonances of X n which are not also resonances of X . • For δ Γ > the statement in Conjecture 1.1 follows from [7, Theorem 1.2], sinceeach resonance s in the half-plane Re( s ) > gives an L -eigenvalue λ = s (1 − s )of the Laplacian. On the other hand, if δ Γ ≤ then X = Γ \ H must be convexcocompact by the results Beardon [1, 2]. Hence the case δ Γ ≤ comes underthe purview of Conjecture 1.1. • We also point out that the aforementioned papers of Brooks [12], Burger [13,14], and Bourgain–Gamburd–Sarnak [7] use solely L -methods to prove uniformspectral gaps. These methods are not available when considering resonancesnot corresponding to eigenvalues.1.2. Statement of results.
The purpose of this paper is to prove some results towardsConjecture 1.1. For the rest of this introduction we fixa Schottky group Γ with Schottky generators S = { γ , . . . , γ m } with m ≥ n of Γ,and we set X = Γ \ H and X n = Γ n \ H . Moreover, for each index n we let G n = Γ / Γ n be the associated covering group, we let p n : Γ → G n be the natural projection map,and we let G n = Cay( G n , S ) be the Cayley graph as in § M n : L ( G n ) → L ( G n )by M n ϕ ( g ) = 1 | S | X s ∈ S ϕ ( p n ( s ) g )for all ϕ ∈ L ( G n ) and g ∈ G n , and we say that the graphs G n form a family of two-sided ε -expanders for some ε > n , all the eigenvalues µ of M n differentfrom 1 satisfy the bound | µ | ≤ − ε . Equivalently, the G n ’s form a family of two-sided ε -expanders if sup n (cid:13)(cid:13)(cid:13) M n | L ( G n ) (cid:13)(cid:13)(cid:13) ≤ − ε, where M n | L ( G n ) denotes the restriction of M n to the subspace L ( G n ) of functions ϕ ∈ L ( G n ) with X x ∈ G n ϕ ( x ) = 0 . This definition coincides with the one contained in Tao’s notes [41]. We note that thenotion of two-sided ε -expanders is very similar to the notion of absolute expanders inKowalski’s book [23]. If we assume that G n is non-bipartite, then the definitions oftwo-sided and absolute expanders coincide. For a discussion on the different notionsof expanders and how they relate to one another, we refer to [23].The first main result is a uniform resonance-free neighbourhood around s = δ Γ underthe assumption that the graphs G n form a family of two-sided ε -expanders. Theorem 1.3.
Assume that the graphs G n form a family of two-sided ε -expanders.Then there exists η > depending only on ε and the Schottky data of Γ such that foreach n , the Selberg zeta function Z Γ n ( s ) has no non-trivial zero with | s − δ Γ | < η . In order to state the next result, we consider the following generalization of the Selbergzeta function: given a finite-dimensional representation ( ρ, V ) of the group Γ, we definethe ρ -twisted Selberg zeta function by the infinite product(2) Z Γ ( s, ρ ) := ∞ Y k =0 Y [ γ ] ∈ [Γ] p det (cid:16) I V − ρ ( γ ) e − ( s + k ) ℓ ( γ ) (cid:17) . We refer to § § Z Γ ( s, ρ ) for representations ρ of G n whose dimension isbounded below by a positive power of the order of G n : Theorem 1.4.
Assume that the graphs G n form a family of two-sided ε -expanders andlet A > and c > be absolute constants. Then there exists η > such that for every n and for every representation ρ of G n with dimension at least c | G n | A , the twistedSelberg zeta function Z Γ ( s, ρ ) has no zeros in the half-plane Re( s ) ≥ δ Γ − η. Moreover, η depends solely on ε , c , A , and the Schottky data of Γ . A direct corollary of Theorem 1.4 and the Venkov–Zograf factorization formula (see § G n ’s form a family of two-sided expanders and the covering groups G n have nosmall non-trivial, irreducible representations: Corollary 1.5.
Assume that the graphs G n form a family of two-sided ε -expanders.Assume furthermore there exist absolute constants A > and c > such that for each n , every non-trivial, irreducible representation of G n has dimension at least c | G n | A .Then there exists η > such that for each n , the Selberg zeta function Z Γ n ( s ) has nonon-trivial zeros in Re( s ) ≥ δ Γ − η . In particular, the surfaces X n have a uniformresonance-free strip, that is, R X n ∩ { Re( s ) ≥ δ Γ − η } = R X ∩ { Re( s ) ≥ δ Γ − η } . for all n .Remark . By an argument in Tao’s notes [42, Proposition 4], we can drop the ex-pander hypothesis in the statement of Corollary 1.5 and replace it with the assumptionthat the random walk on G n is “flattening” in the following sense: there exist constants0 < β < A and C > n there exists some N ≤ C log | G n | ,(3) 1 | S | N { ( γ i , . . . , γ i N ) ∈ S N : p n ( γ i · · · γ i N ) = id G n } ≤ C | G n | − / β . In fact, (3) together with the assumption that each non-trivial irreducible representa-tion of G n has dimension at least c | G n | A for some absolute constants c > A > G n are two-sided ε -expanders for some ε > c, C, A, β. By the work of Landazuri–Seitz [43], we know that for every finite simple group ofLie type G , each non-trivial representation of G has dimension at least c | G | A forsome absolute constants c > A >
0. On the other hand, from the work ofBreuillard–Green–Guralnick–Tao [11] (which builds on a number of previous papers),we know that simple groups of Lie type give rise to expanders. This allows us touse Corollary 1.5 in conjunction with [11] to produce explicit families of covers withuniform resonance-free strips. Let us be more precise. We fix a non-elementary convexcocompact base surface X = Γ \ H where Γ is freely generated by m ≥ γ ± , . . . , γ ± m , and we fix a finite simple group of Lie type G . Now we chooseelements g , . . . , g m in G independently and uniformly at random and we let φ : Γ → G L. SOARES be the homomorphism defined by φ ( γ j ) = g j for all j ∈ { , . . . , m } . Finally, we setΓ φ = ker( φ ) and we let X φ = Γ φ \ H be the associated random cover of X . Noticethat if the elements g , . . . , g m generate the whole group G , then X φ is a G -cover of X (that is, the covering group Γ / Γ φ is isomorphic to G ). From the main result in [11] ,we deduce that there exists a constant ε > G , suchthat with probability tending to 1 as | G | → ∞ , the elements g , . . . , g m generate thegroup G and the Cayley graph G = Cay( G, { g ± , . . . , g ± m } )is a two-sided ε -expander. Combining this fact with Corollary 1.5, we obtain thefollowing result, which roughly speaking says that almost all G -covers arising from thedescription above have a uniform resonance-free strip. Corollary 1.7.
Let X be a non-elementary convex cocompact hyperbolic surface andlet r ≥ be an integer. Then there exists η > depending only on r and X such thatfor every finite simple group G of Lie type of rank r , the probability that the surface X φ is a G -cover of X with R X φ ∩ { Re( s ) ≥ δ Γ − η } = R X ∩ { Re( s ) ≥ δ Γ − η } tends to as | G | → ∞ .Remark . We note that this result fails if G abelian. In fact, we know from [22] thatlarge abelian covers of X have an abundance of resonances arbitrarily close to s = δ Γ .We also note that Corollary 1.7 is similar in spirit to the result of Naud–Magee [29]which says that random coverings of X of degree n have a uniform resonance-free stripwith probability tending to 1 as n → ∞ .Another consequence of Theorem 1.4 (although less direct) and one of its most inter-esting applications is a uniform resonance-free strip for the family of congruence coversof convex cocompact hyperbolic manifolds, which was recently proven by Oh–Winter[31]. We recall that for every group Γ ⊂ SL ( Z ) and for every integer q ≥ q of Γ is given byΓ( q ) := { γ ∈ Γ : γ ≡ I (mod q ) } . In this paper we give a new and more direct proof of the following result based onTheorem 1.4, the work on expanders by Bourgain–Varj´u [8] and an inductive argumenton the sum of the exponents of the prime factors of q . Theorem 1.9.
Let Γ ⊂ SL ( Z ) be a convex cocompact Fuchsian group. Then thereexists an integer q = q (Γ) and η = η (Γ) > such that for all integers q ≥ with ( q, q ) = 1 the Selberg zeta function Z Γ( q ) ( s ) has no non-trivial zeros in Re( s ) ≥ δ Γ − η and R X ( q ) ∩ { Re( s ) ≥ δ Γ − η } = R X ∩ { Re( s ) ≥ δ Γ − η } , where X ( q ) = Γ( q ) \ H is the corresponding congruence cover. Here, ( a, b ) denotes thelargest common divisor of a and b .Remark . The statement of Theorem 1.9 is true for all non-elementary, finitelygenerated Fuchsian groups Γ ⊂ SL ( Z ). For the modular group Γ = SL ( Z ) (in whichcase we have δ Γ = 1) Selberg’s famous -theorem [40] gives a more precise statementwith the explicit spectral gap η = . For δ Γ > , Theorem 1.9 follows from Gamburd’sthesis [17] with η = δ Γ − . For δ Γ > , the statement was proven by Bourgain–Gamburd–Sarnak [7]. The more difficult case δ Γ ≤ is covered by Theorem 1.9, as See also the remark after Theorem 2.3 in [11] explained in Remark 1.2. We also point out that our proof (similarly to the proof ofOh–Winter [31]) does not yield any explicit bounds on the spectral gap η , since it relieson the expander theory developed by Bourgain–Gamburd [6], Bourgain–Varj´u [8] andothers, which does not provide explicit expansion constants.1.3. Organization and overview of proofs.
Many aspects of our proofs are similarto those contained in the work on congruence covers by Bourgain–Gamburd–Sarnak[7] and Oh–Winter [31]. However, their methods do not seem to generalize easily tothe more general situation considered in this paper, so we have been forced to devisenew arguments. The aim of this subsection is to give an overview of the main ideas,without delving into technical details or elaborating on the notations.In § . (b) Twisted Selberg zeta functions Z Γ ( s, ρ ) and their realization as Fredholm de-terminants of transfer operators. The family of transfer operators in question,denoted by L s,ρ , is parametrized by the spectral variable s ∈ C and they acton vector-valued Bergman spaces.(c) The Venkov–Zograf factorization formula, according to which the Selberg zetafunction of the subgroup Γ n < Γ is the product of twisted Selberg zeta functions Z Γ ( s, ρ ) over all the irreducible representations ρ of G n = Γ / Γ n In § Γ , sometimes called the Bowen–Series map, whose properties we briefly recall in §
4. We record some properties of the topological pressure associated to this dynamicalsystem and the Ruelle–Perron–Frobenius theorem. Moreover, we develop some a prioriestimates for the transfer operators acting on vector-valued Bergman spaces.In § G n to establish estimates forcertain sums over words W N of length N (Proposition 5.1). As we will explain in § { , . . . , m } , where m is the number of Schottky generators of Γ. Letting γ a denote the element in Γ cor-responding to the word a , we show that for each non-trivial irreducible representation( ρ, V ) of G n , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W N ρ ( γ a ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) End( V ) ≪ e − cN | W N | , where c > n , N and ρ. We then use a decoupling argument to generalizethis purely combinatorial statement to more general sums over words of length N ,where the summands are now weighted by functions g : Λ Γ → V on the limit set Λ Γ (Proposition 5.3). More precisely, we prove(4) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W N ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≪ e − cN | W N | · N ( g )for all x ∈ Λ Γ , where N ( · ) is a certain norm defined for functions g : Λ Γ → V .In § ρ -twisted transfer operator at s = δ Γ (Proposition L. SOARES kL Nδ Γ ,ρ f k ≪ e − cN k f k for all functions f belonging to the V -valued Bergman space. This improves upon the“trivial” estimate kL Nδ Γ ,ρ f k ≪ k f k . The key feature of (5) is that the exponential decay rate c > ρ, V ) ranging over the non-trivial irreducible representations of the groups G n ,under the assumption that the graphs G n form a family of two-sided expanders.In § § η Γ > T Γ > X ′ of X = Γ \ H , there are no resonances s for X ′ with Re( s ) > δ Γ − η Γ and | Im( s ) | > T Γ . Thus, by the Venkov–Zograf factorizationformula, proving Theorem 1.4 amounts to proving that the function Z Γ ( s, ρ ) has nozeros in(6) R η ,T Γ := { Re( s ) ≥ δ Γ − η , | Im( s ) | < T Γ } for some uniform η > . Thanks to the transfer operator theoretic point of view ofthe twisted Selberg zeta function (Proposition 2.1), we know that Z Γ ( s, ρ ) vanishes at s ∈ C if and only if there exists some non-zero function f satisfying f = L s,ρ f. Hence, to exclude the existence of zeros in (6) it suffices to show that for some suitablepositive integer N , the norm of L Ns,ρ is strictly less than 1 for all s ∈ R η ,T Γ . Our strat-egy is to obtain an estimate on the norm L Ns,ρ via an estimate on the Hilbert–Schmidtnorm k · k HS for the operator L Ns,R n , where R n denotes the regular representation ofthe group G n . Combining the flattening result mentioned above with some distortionestimates for the derivatives of γ a , we obtain for N ≈ log | G n | , kL Ns,R n k ≪ t | G n | C ( δ − σ ) , where σ and t are the real and imaginary parts of s , respectively. Here C > G n ’s and theSchottky data of Γ. We then argue that for each irreducible representation ( ρ, V ) of G n , dim( ρ ) kL Ns,ρ k ≤ kL Ns,R n k , where dim( ρ ) = dim( V ) is the dimension of ρ . We can finally make use of the assump-tion that the dimension of ρ is bounded below polynomially in terms of the size of G n :inserting dim( ρ ) ≫ | G n | A into the above estimates yields kL Ns,ρ k ≪ t | G n | C ( δ Γ − σ − AC ) . Thus, for σ ≥ δ Γ − A C we obtain kL Ns,ρ k ≪ t | G n | − A/ . Hence, setting η = A C we obtain for all s ∈ R η ,T Γ (assuming that G n is sufficientlylarge), kL Ns,ρ k < , as desired.1.4. Notation.
We write f ( x ) ≪ g ( x ) or f ( x ) = O ( g ( x )) interchangeably to meanthat there exists an implied constant C > | f ( x ) | ≤ C | g ( x ) | for all x ≥ C ,and we write f ( x ) ≪ y g ( x ) or f ( x ) = O y ( g ( x )) to indicate that C depends on y . Also,we write f ( x ) ≍ g ( x ) to mean g ( x ) ≪ f ( x ) ≪ g ( x ). For a finite set S we denote by | S | its cardinality. Additional notations will be explained as they are introduced.2. Preliminaries
Schottky groups.
In this section we briefly review some properties of Schottkygroups, referring the reader to Borthwick’s book [3, §
15] for a comprehensive discussion.The group SL ( R ) acts on the extended complex plane C = C ∪ {∞} by M¨obiustransformations γ = (cid:18) a bc d (cid:19) ∈ SL ( R ) , z ∈ C = ⇒ γ ( z ) = az + bcz + d . A Schottky group is a convex cocompact subgroup Γ ⊂ SL ( R ) constructed in thefollowing way: • Fix m ≥ D , . . . , D m ⊂ C with centerson the real line, • For every j ∈ { , . . . , m } let γ j ∈ SL ( R ) be the isometry that maps the exteriorof D j to the interior of D j + m , for every j ∈ { m + 1 , . . . , m } set γ m − j := γ − j and extend this definition cyclically to all j ∈ Z by setting γ j := γ j mod 2 m . • Let Γ ⊂ SL ( R ) be the free group generated by the elements γ , . . . , γ m . One of the standard models for the hyperbolic plane is the Poincar´e half-plane H = { x + iy ∈ C : y > } endowed with its standard metric of constant curvature − ds = dx + dy y . By Button’s result [15], every convex cocompact hyperbolic surface X can be realizedas the quotient of H by a Schottky group Γ, see also [3, Theorem 15.3]. Notice thatthe complement H r S mj =1 D j provides a fundamental domain for the action of Γ on H . Given a convex cocompact hyperbolic surface X we fix a Schottky group Γ suchthat X ∼ = Γ \ H and we refer to D , . . . , D m and γ , . . . , γ m as the Schottky data of X (or Γ) and we refer to γ , . . . , γ m as the Schottky generators of Γ. ∂ H H D D D D D D γ γ γ Figure 1.
A configuration of Schottky disks and isometries with m = 32.2. Selberg zeta function and transfer operator.
In this paper, twisted
Selbergzeta functions play a crucial role. Let Γ < SL ( R ) be a finitely generated Fuchsiangroup. As is well-known, the set of prime periodic geodesics on X = Γ \ H is bijectiveto the set [Γ] p of Γ-conjugacy classes of the primitive hyperbolic elements in Γ (see forinstance [3]). We denote by ℓ ( γ ) the length of the geodesic corresponding to [ γ ] ∈ [Γ] p .Let V be finite-dimensional complex vector space endowed with a hermitian innerproduct h· , ·i V and induced norm k v k V = p h v, v i V , and let ρ : Γ → U( V ) be a unitaryrepresentation of Γ. The twisted Selberg function is then defined for Re( s ) ≫ Z Γ ( s, ρ ) := ∞ Y k =0 Y [ γ ] ∈ [Γ] p det (cid:16) I V − ρ ( γ ) e − ( s + k ) ℓ ( γ ) (cid:17) . Notice that (7) reduces to the classical Selberg zeta function when ρ is the trivialone-dimensional representation of Γ . Now fix a Schottky group Γ and let D , . . . , D m and γ , . . . , γ m be the correspondingSchottky data, which we assume to be fixed, and let D be the union of the disks D j , D := m [ j =1 D j . Consider the Hilbert space H ( D, V ) of square-integrable, holomorphic functions on D ,(8) H ( D, V ) := { f : D → V holomorphic | k f k < ∞ } , where k f k is the L -norm k f k := Z D k f ( z ) k V dvol( z ) . Here, vol denotes the Lebesgue measure. Notice that for V = C , (8) reduces to theclassical Bergman space H ( D ). Finally, we define the twisted transfer operator(9) L s,ρ : H ( D, V ) → H ( D, V )by the formula(10) L s,ρ f ( z ) = m X i =1 i = j γ ′ i ( z ) s ρ ( γ i ) − f ( γ i ( z )) if z ∈ D j for all f ∈ H ( D, V ). The twisted Selberg zeta function is related to the transferoperator L s,ρ through the following result: Proposition 2.1 (Fredholm determinant identity) . For every s ∈ C the operator (9) is trace class and we have the identity (11) Z Γ ( s, ρ ) = det(1 − L s,ρ ) . Identities such as (11) are well-known in thermodynamic formalism, a subject goingback to Ruelle [38], at least in the case where ρ is the trivial one-dimensional repre-sentation. The relation between the Selberg zeta function and transfer operators hasbeen studied by a number of different authors. For the convex cocompact setting (nocusps) we refer to [36, 37, 18]. The extension to non-trivial twists ρ can be found inthe more recent papers [16, 35, 30, 29]. A proof of Proposition 2.1 can be found in[22].Proposition 2.1 has many remarkable corollaries. In particular, since L s,ρ dependsholomorphically on the variable s , it follows directly that the twisted Selberg function Z Γ ( s, ρ ) is an entire function when Γ is a Schottky group, which is far from obviousfrom its definition in (7) as an infinite product over conjugacy classes.2.3. Finite covers and Venkov–Zograf formula.
Twisted Selberg zeta functionsare extremely helpful when studying resonances in families of covers for hyperbolicsurfaces. To see why this is the case, let X = Γ \ H be a geometrically finite hyperbolicsurface and let X ′ = Γ ′ \ H be a regular covering of X of finite degree. We mayassume without loss of generality that Γ ′ is a normal subgroup of Γ and we denoteby G = Γ / Γ ′ the associated covering group (sometimes also called the Galois group ofthe covering). The (left) regular representation R of the group G is the representation R : G → U( L ( G )) defined by R ( g ) ϕ ( x ) = ϕ ( g − x ) . for all g ∈ G , x ∈ G and ϕ ∈ L ( G ). This representation can be extended in a naturalway to a representation R : Γ → U( L ( G )) by setting(12) R ( γ ) := R ( p ( γ ))for all γ ∈ Γ, where p : Γ → G is the natural projection map. Then we have the relation(13) Z Γ ′ ( s ) = Z Γ ( s, R ) , which was proven by Venkov–Zograf [45] in the case where Γ is a cofinite Fuchsiangroup (see also [44]). For an extension of this formula to the non-cofinite case we referto [16].Twisted Selberg functions enjoy the following nice property: for any two finite-dimensionalunitary representations ( ρ , V ) and ( ρ , V ) of Γ, we have the factorization(14) Z Γ ( s, ρ ⊕ ρ ) = Z Γ ( s, ρ ) Z Γ ( s, ρ ) , where the symbol ⊕ stands for the direct sum of representations. As is well-known,the regular representation decomposes as the direct sum(15) R = M ( ρ,V ) ∈ b G dim( ρ ) ρ, where b G denotes the finite set of all irreducible representations ( ρ, V ) of Γ (up toequivalence of representations). Hence, keeping in mind that each representation ρ of G extends to a representation of Γ similarly to (12), we obtain the formula(16) Z Γ ′ ( s ) = Y ρ ∈ b G Z Γ ( s, ρ ) dim( ρ ) , which we will refer to as the Venkov–Zograf (factorization) formula.3. Deducing Theorem 1.9 from Theorem 1.4
In this section, we deduce Theorem 1.9 from Theorem 1.4. Let Γ ⊂ SL ( Z ) be a convexcocompact Fuchsian group and let S ⊂ SL ( Z ) denote the set of Schottky generatorsof Γ . Recall that for every integer q ≥ q congruence subgroup Γ( q ) of Γ isdefined as the kernel of the reduction modulo q map π q : Γ → G q := SL( Z /q Z ) , γ γ (mod q ) . The main input for our proof is the deep result of Bourgain–Varj´u [8] which implies thatthe Cayley graphs Cay( π q (Γ) , π q ( S )) form a family of two-sided ε -expanders, where q runs through the positive integers. Moreover, there is an integer q depending only onthe set S of generators such that π q (Γ) = G q for all q with ( q, q ) = 1. In particular,the covering group Γ / Γ( q ) is isomorphic to G q provided ( q, q ) = 1.By the Venkov–Zograf formula (16), the statement of Theorem 1.9 is equivalent to theexistence of some η > q ≥ q, q ) = 1 and each non-trivialrepresentation ρ of G q , the Selberg zeta function Z Γ ( s, ρ ) has no zeros in Re( s ) ≥ δ Γ − η. We will use the following result pertaining to the representation theory of the groups G q . Lemma 3.1 (Dichotomy for representations of G q ) . For all integers q ≥ and for eachnon-trivial irreducible representation ρ of G q one of the following statements holds: (i) the dimension of ρ is bounded below by c | G q | / for some absolute constant c > , or (ii) ρ “descends” to a representation ρ ′ of G q ′ with q ′ | q , q ′ < q . More precisely, ρ isequivalent to the tensor product of representations ρ ′ ⊗ id V where id V denotesthe trivial representation on some vector space V .Here, we write q ′ | q to mean that q ′ is a divisor of q . Before we prove Lemma 3.1, let us see how it can be used in conjunction with Theorem1.4 to deduce Theorem 1.9. Let c > η > q ≥ q, q ) = 1 and for each non-trivial representation ρ of G q whose dimension is bounded below by c | G q | / , the twisted Selberg zeta function Z Γ ( s, ρ ) has no zeros in Re( s ) ≥ δ Γ − η .We will argue by induction on the sum of the powers of the prime factors of q = p k · · · p k r r , ℓ ( q ) = k + · · · + k r . If ℓ ( q ) = 1 then q is a prime number. By Lemma 3.1 the non-trivial representations ρ of G q satisfy the bound dim( ρ ) ≥ c | G q | / and thus Z Γ ( s, ρ ) has no zeros in Re( s ) ≥ δ Γ − η .Hence, the statement of Theorem 1.5 is true in the case ℓ ( q ) = 1.Now let q be a positive integer with ℓ ( q ) = ℓ with ℓ >
1, and assume the statement ofTheorem 1.9 is known to be true with η chosen as above for all positive integers q ′ with( q ′ , q ) = 1 and ℓ ( q ′ ) < ℓ . Then by Lemma 3.1 we have either dim( ρ ) > c | G q | / , inwhich case we already know that Z ( s, ρ ) has no zeros in Re( s ) ≥ δ Γ − η , or ρ descendsto a representation ρ ′ of G q ′ with q ′ | q and q ′ < q . In the latter case we clearly have ℓ ( q ′ ) < ℓ . But the zeros of Z Γ ( s, ρ ) coincide with the zeros of Z Γ ( s, ρ ′ ) and the resultfollows by induction. Hence, the proof of Theorem 1.9 is finished, assuming Theorem1.3 and Lemma 3.1. The latter will be proven here. Proof of Lemma 3.1.
Assume q is prime. By a classical result due to Frobenius, everynon-trivial representation of G q has dimension at least q − . Observe that since | G q | ≍ q , we can bound the dimension of each non-trivial representation of G q from belowby c | G q | / for some small absolute constant c > q > q = p k · · · p k r r be the prime factorization of q . Then G q can be written as the direct product(17) G q ∼ = r Y j =1 G p kjj and each irreducible representation ρ of G q can be written as(18) ρ = ρ ⊗ · · · ⊗ ρ r where each ρ j is an irreducible representation of G p kjj .For each j = 1 , . . . , r suppose the following: if k j = 1 then ρ j is non-trivial and if k j > ρ j is faithful. (Recall that a representation ( ρ, V ) of a group G is called faithful if ρ : G → U( V ) is an injective homomorphism). Then, by a result of Bourgain–Gamburd[5, Lemma 7.1], we havedim( ρ j ) ≥ ( p k j − j ( p −
1) if k j ≥ ( p −
1) if k j = 1 . Thus, using the bound dim( ρ j ) ≥ p k j j which is valid in both cases, we obtain a lowerbound on the dimension of ρ ,dim( ρ ) ≥ r Y j =1 dim( ρ j ) ≥ r Y j =1 p k j j = 3 − r q ≫ ε q − ε for all ε >
0. Recalling that the size G q is given by | G q | = q r Y j =1 − p j ! < q , we deduce that dim( ρ ) ≫ ε | G q | / − ε ≥ c | G q | / for some sufficiently small absolute constant c > j such that k j = 1 and ρ j is the trivial represen-tation. Then ρ is equivalent to ρ ′ = ρ ⊗ · · · ⊗ ρ j − ⊗ id ⊗ ρ j +1 ⊗ · · · ⊗ ρ r , thus ρ descends to a representation of G q ′ where q ′ = qp j .Now suppose that there exists j such that k j > ρ j is not faithful. Then ker( ρ j )is a proper normal subgroup of G p kjj . Given a prime p and integers 1 ≤ k ′ < k , let H pk,k ′ denote the kernel of the surjective homomorphism π pk,k ′ : G p k → G p k ′ The set { H pk,k ′ : 1 ≤ k ′ ≤ k } is a complete list of the normal subgroups of G p k up toisomorphism. Hence ker( ρ j ) is isomorphic to H p j k j ,k ′ j for some 1 ≤ k ′ j < k j . Thus ρ j descends to a homomorphism ρ ′ j : G p kjj /H p j k j ,k ′ j ∼ = G p k ′ jj → U( V ) . Consequently, ρ descends to the representation ρ ′ = ρ ⊗ · · · ⊗ ρ ′ j ⊗ · · · ⊗ ρ r of G q ′ , where q ′ = p k · · · p k ′ j j · · · p k r r . Noticing that q ′ | q is a proper divisor of q completesthe proof of Lemma 3.1. (cid:3) A priori estimates
Reduced words and bounds for derivatives.
From now on, we assume that X = Γ \ H is a fixed convex cocompact hyperbolic surface, and we assume that Γ is aSchottky group with Schottky data D , . . . , D m and γ , . . . , γ m . We will henceforthdrop Γ from the subscripts, writing δ = δ Γ and Λ = Λ Γ . Moreover, we denote by L s,ρ the transfer operator introduced in § a = a · · · a N with a , . . . , a N ∈ { , . . . , m } , we set γ a := γ a ◦ · · · ◦ γ a n Recall that Γ is freely generated by the elements γ , . . . , γ m , that is, there are no rela-tions among the elements γ , . . . , γ m except for the trivial relations γ − i γ i = γ i γ − i = e .Recall also that γ − i = γ i + m , where the indices are defined modulo 2 m . We say thatthe word a is reduced if the corresponding element γ a is reduced when viewed as aword in the alphabet { γ , . . . , γ m } . We then denote by W N the set of reduced wordsof length N ,(19) W N = { a = a · · · a N : a i = a i +1 + m (mod 2 m ) for all i = 1 , . . . , N − } . Notice that the set W N corresponds one-to-one to the elements of reduced word length N in Γ via the map a γ a . Moreover, we define for all j = 1 , . . . , m the set of words W jN = { a = a · · · a N ∈ W N : a N = j } . Observe that if a = a · · · a N ∈ W jN then γ a maps the closure of D j into the interior D a + m . With these notations in place, we have for all N ≥ j ∈ { , . . . , m } the formula(20) L Ns,ρ f ( z ) = X a ∈ W jN γ ′ a ( z ) s ρ ( γ a ) − f ( γ a ( z )) , if z ∈ D j . Finally, we record some distortion estimates (see [27] for the proofs) which are crucialin this work: • (Uniform hyperbolicity) There exist constants c , c > < θ < θ < N , all j ∈ { , . . . , m } , and all a ∈ W jN , we have(21) c θ N < sup z ∈ D j | γ ′ a ( z ) | < c θ N . • (Bounded distortion) There exists a constant c > N , all j ∈ { , . . . , m } , all a ∈ W jN , and all z ∈ D j we have(22) sup z ∈ D j (cid:12)(cid:12)(cid:12)(cid:12) γ ′′ a ( z ) γ ′ a ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c • (Bounded distortion for third derivatives) There exists a constant c > N , all j ∈ { , . . . , m } , all a ∈ W jN , and all z ∈ D j , we havesup z ∈ D j (cid:12)(cid:12)(cid:12)(cid:12) γ ′′′ a ( z ) γ ′ a ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c The bounded distortion estimate has the following important consequence: there existsa constant c > a ∈ W jN and all pair of points z , z ∈ D j ,we have(23) (cid:12)(cid:12)(cid:12)(cid:12) γ ′ a ( z ) γ ′ a ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c . Topological pressure and Ruelle-Perron-Frobenius theorem.
Letting D , . . . , D m and γ , . . . , γ m denote the Schottky data of the group Γ, we define for each j ∈{ , . . . , m } the interval I j := D j ∩ R . Notice that the I j ’s are mutually disjoint realintervals. Let I = S mj =1 I j denote its union. It turns out that the map T : I → I given by T ( x ) = γ j ( x ) if x ∈ I j , sometimes called the Bowen–Series map , encodes the dynamics of the full group Γ.The origins of this type of coding go back to the work of Bowen–Series [10]. The limitset Λ can be re-interpreted as the non-wandering set of the map T as(24) Λ = ∞ \ N =1 T − N ( I ) . Given any continuous function ϕ : I → R , we define the topological pressure in termsof weighted sums over periodic orbits through the formula(25) P ( ϕ ) = lim N →∞ N log X T N x = x e ϕ ( n ) ( x ) ! where ϕ ( n ) ( x ) := ϕ ( x ) + ϕ ( T x ) + . . . + ϕ ( T n − x ) . Another definition of P ( ϕ ) is by thevariational formula(26) P ( ϕ ) = sup µ (cid:18) h µ ( T ) − Z Λ ϕdµ (cid:19) , where µ ranges over the set of T -invariant probability measures, and h µ ( T ) is themeasure theoretic entropy. We refer the reader to [32] for general facts on topologicalpressure and thermodynamical formalism. More important for us is Bowen’s celebratedresult [9] which says that the map(27) R → R , σ P ( σ ) := P ( − σ log | T ′ | )is convex, strictly decreasing and vanishes precisely at σ = δ , the Hausdorff dimensionof the limit set. The relevance of the topological pressure stems from the Ruelle-Perron-Frobenius theorem: Proposition 4.1 (Ruelle-Perron-Frobenius) . Set L σ = L σ, id where σ ∈ R is real and id is the one-dimensional trivial representation. Then the following statements holdtrue: • The spectral radius of L σ on C ( I ) is e P ( σ ) which is a simple eigenvalue asso-ciated to a strictly positive eigenfunction ϕ σ > in C ( I ) . • The operator L σ on C ( I ) is quasi-compact with essential spectral radius smallerthan κ ( σ ) e P ( σ ) for some κ ( σ ) < . • There are no other eigenvalues on | z | = e P ( σ ) . • Moreover, the spectral projector P σ on { e P ( σ ) } is given by P σ ( f ) = ϕ σ Z Λ f dµ σ , where µ σ is the unique T -invariant probability measure on Λ that satisfies L ∗ σ ( µ σ ) = e P ( σ ) µ σ . For a proof, we refer to [32, Theorem 2.2] (see also [22, Proposition 4.4]). One conse-quence of the Ruelle–Perron–Frobenius theorem is the following estimate due to Naud[28], which we will use multiple times in subsequent sections of this paper:
Lemma 4.2 (Pressure estimate) . For every
M > there exists a constant C > suchthat for all σ ∈ [0 , M ] and all N ≥ we have (28) m X j =1 X a ∈ W jN k γ ′ a k σ ∞ ,D j ≤ Ce NP ( σ ) , where we use the notation k g k ∞ ,D j := sup z ∈ D j | g ( z ) | , and P ( σ ) is the pressure function defined in (27) . Bergman space estimates.
In this subsection we develop some estimates forfunctions belonging to the Bergman space H ( D, V ). Recall that D is the union of theSchottky disks D , . . . , D m of Γ, V is a finite-dimensional vector space endowed witha hermitian inner product h· , ·i V and induced norm k v k V = p h v, v i V , and H ( D, V )is the V -valued Bergman space defined in (8).Working with Bergman spaces has the following advantage: for every function f ∈ H ( D, V ) and every point z ∈ D bounded away from the boundary ∂D , the sizes ofboth f ( z ) and the derivative f ′ ( z ) are controlled by the L -norm of f , which is givenby(29) k f k := Z D k f ( z ) k V dvol( z ) . More precisely, we have
Lemma 4.3 (Bergman space estimates) . For every ε > there exists a constant C > such that for every point z ∈ D with dist( z, ∂D ) ≥ ε and every f ∈ H ( D, V ) , wehave (30) k f ( z ) k V ≤ C k f k and (31) k f ′ ( z ) k V ≤ C k f k . Here, dist denotes the euclidean distance dist( z, A ) = inf w ∈ A | z − w | . Remark . For any given basis e , . . . , e d of the vector space V we can write thefunction f ∈ H ( D, V ) as(32) f ( z ) = f ( z ) e + · · · + f d ( z ) e d where each f k belongs to the classical Bergman space H ( D ). We can then define thederivative of f componentwise, that is,(33) f ′ ( z ) = f ′ ( z ) e + · · · + f ′ d ( z ) e d . Proof.
Let f ∈ H ( D, V ). Let d = dim( V ) be the dimension of V , fix an orthonormalbasis e , . . . , e d for V , and decompose f as in (32). Then we have k f ( z ) k V = d X k =1 k f k ( z ) k V , k f ′ ( z ) k V = d X k =1 k f ′ k ( z ) k V for all z ∈ D and k f k = d X k =1 k f k k , so we can assume without loss of generality that d = 1 and f ∈ H ( D ).Now let z ∈ D be a point with dist( ∂D, z ) ≥ ε . Then we have D ( z , ε ) ⊂ D , where D ( z , r ) denotes the open disk in the complex plane with center z and radius r . Since f is holomorphic on D , we have f ( z ) = 1vol( D ( z , ε )) Z D ( z , ε ) f ( z ) dvol( z ) . Applying the Cauchy–Schwarz inequality yields | f ( z ) | ≤ D ( z , ε )) Z D ( z , ε ) | f ( z ) | dvol( z ) ≤ D ( z , ε )) k f k . Using vol( D ( z , ε )) = π (cid:0) ε (cid:1) ≥ (cid:0) ε (cid:1) , this implies | f ( z ) | ≤ ε − k f k , proving (30). To prove (31), we use the Cauchy integral formula f ′ ( z ) = 12 πi Z ∂D ( z , ε ) f ( z )( z − z ) d z to estimate(34) | f ′ ( z ) | ≤ ε − max w ∈ ∂D ( z , ε ) | f ( w ) | . Notice that w ∈ ∂D ( z , ε ) implies dist( ∂D, w ) ≥ ε . Hence, combining (30) with (34)yields | f ′ ( z ) | ≤ ε − k f k , completing the proof of Lemma 4.3. (cid:3) Lemma 4.5.
There exists
C > such that for all f ∈ H ( D, V ) we have k f k ∞ , Λ ≤ C k f k , where k f k ∞ , Λ := sup x ∈ Λ k f ( z ) k V . Proof.
By Lemma 4.3, it suffices to show that each point in the limit set is uniformlybounded away from the boundary ∂D.
To see this we use (24) to rewrite the limit setin terms of reduced words as(35) Λ = \ N ≥ m [ j =1 [ a ∈ W jN γ a ( I j ) , where I j = D j ∩ R is the interval spanned by the Schottky disk D j . For all j andfor every a = a . . . a N ∈ W jN , the map γ a maps the closure of I j into the interior of I a + m . Thus, there is a constant c > γ a ( I j ) , ∂D ) = dist( γ a ( I j ) , ∂I a + m ) ≥ c. Using (35), we deduce that for any point x ∈ Λ in the limit set we have dist( x, ∂D ) ≥ c .The claim then follows from Lemma 4.3. (cid:3) Lemma 4.6 (mean value estimate) . There exists a constant
C > such that foreach j ∈ { , . . . , m } , N ≥ , a ∈ W jN , all points z, z ′ ∈ D j , and each function f ∈ H ( D, V ) , we have (36) k f ( γ a ( z )) − f ( γ a ( z ′ )) k V ≤ C k γ ′ a k D j , ∞ k f k and (37) X a ∈ W jN k f ( γ a ( z )) − f ( γ a ( z ′ )) k V ≤ Ce P (1) N k f k , where P ( · ) is the pressure defined in (27) .Proof. Notice that the second estimate follows from the first one and the pressureestimate in Lemma 4.2. To prove the first estimate, we use the same argument as inthe proof of Lemma 4.3 to reduce the proof to the one dimensional case. Thus weassume that V = C and f ∈ H ( D ). Fix points z, z ′ ∈ D . By the mean value theoremthere exists a point z ′′ on the line segment connecting z and z ′ such that f ( γ a ( z )) − f ( γ a ( z ′ )) = γ ′ a ( z ′′ ) f ′ ( γ a ( z ′′ ))( z − z ′ ) . We deduce that | f ( γ a ( z )) − f ( γ a ( z ′ )) | ≤ k γ ′ a k ∞ ,D j diam( D j ) | f ′ ( γ a ( z ′′ )) | . Notice that γ a ( z ′′ ) is bounded away from the ∂D , that is, there exists c > j, N, a , z, z ′ such that dist( γ a ( z ′′ ) , ∂D ) ≥ c. Applying Lemma 4.3 gives | f ( γ a ( z )) − f ( γ a ( z ′ )) | ≤ C k γ ′ a k ∞ ,D j k f k for some uniform constant C >
0, completing the proof. (cid:3) A priori estimates for the transfer operator.
In this section we prove apriori estimates for the norm of L Nδ,ρ in the space H ( D, V ). Recall that for everyfunction f : D → V , we set k f k ∞ , Λ := sup x ∈ Λ k f ( x ) k V , whereas k f k denotes the L -norm defined in (29). The next estimate, which will becrucial at a later stage of this work, says that the L -norm of L Nδ,ρ f is essentiallycontrolled by the maximum norm of f on the limit set. Lemma 4.7 (A priori estimates) . There exist constants c > and C > dependingonly on the Schottky data of Γ such that for every finite-dimensional unitary represen-tation ( ρ, V ) of Γ , all f ∈ H ( D, V ) , and all N , (38) kL Nδ,ρ f k ≤ C (cid:0) k f k ∞ , Λ + e − cN k f k (cid:1) . In particular, there exists
C > depending only on the Schottky data of Γ such that (39) kL Nδ,ρ f k ≤ C k f k . Proof.
The second estimate follows from the first one and Lemma 4.5. To prove (38)fix j ∈ { . . . , m } and a point z ∈ D j , and recall from (20) that L Nδ,ρ f ( z ) = X a ∈ W jN γ ′ a ( z ) δ ρ ( γ a ) − f ( γ a ( z )) . Now fix a point x ∈ Λ ∩ D j and write L Nδ,ρ f ( z ) = X a ∈ W jN γ ′ a ( z ) δ ρ ( γ a ) − f ( γ a ( x )) + X a ∈ W jN γ ′ a ( z ) δ ρ ( γ a ) − ( f ( γ a ( z )) − f ( γ a ( x ))) . The limit set Λ is invariant under the action of Γ, so we have γ a ( x ) ∈ Λ for all words a . Thus, using the triangle inequality (together with the fact that ρ is unitary) andthe mean value estimate in Lemma 4.6, we obtain kL Nδ,ρ f ( z ) k V ≤ X a ∈ W jN k γ ′ a k δ ∞ ,D j k f ( γ a ( x )) k V + X a ∈ W jN k γ ′ a k δ ∞ ,D j k f ( γ a ( z )) − f ( γ a ( x )) k≪ X a ∈ W jN k γ ′ a k δ ∞ ,D j k f k ∞ , Λ + X a ∈ W jN k γ ′ a k δ +1 ∞ ,D j k f k . Invoking the pressure estimate in Lemma 4.2, this gives kL Nδ,ρ f ( z ) k V ≪ k f k ∞ , Λ + e P ( δ +1) N k f k . where the implied constant depends only on the Schottky data of Γ. Integrating thisestimate over all z ∈ D yields kL Nδ,ρ f k = Z D kL Nδ,ρ f ( z ) k dvol( z ) ≪ k f k ∞ , Λ + e P ( δ +1) N k f k . This proves (38) with c = − P ( δ + 1) > (cid:3) Exploiting the expansion property
Exponential decay estimate for W N . Let us briefly recall some notations.Given a family Γ n of finite-index, normal subgroups of Γ, we let G n = Γ / Γ n be the(finite) quotient groups and we let p n : Γ → G n be the natural projection maps. Wewill always assume that Γ is a Schottky group as in § S = { γ , . . . , γ m } be the set of Schottky generators of Γ. Moreover, we assume that Γ is non-elementary,which is equivalent with saying that m ≥
2. We then define the family of Cayleygraphs G n = Cay( G n , S )as in § G n are said to form a family of two-sided ε -expandersif there exists a uniform ε > µ with µ = 1 of theMarkov operator(40) M n : L ( G n ) → L ( G n ) , M n ϕ ( x ) = 1 | S | X s ∈ S ϕ ( p n ( s ) x )satisfy | µ | ≤ − ε. We denote by b G n the finite set of irreducible representations of G n and we let b G n = b G n r { id } denote the set of non-trivial irreducible representations of G n . It is importantto notice that given any representation ( ρ, V ) of G n , we can always endow the repre-sentation space V with a hermitian inner product h· , ·i V with respect to which ρ ( g ) isunitary for every g ∈ G n . We will always tacitly assume that V is endowed with suchan inner product and we let k v k V = p h v, v i V denote the induced norm. Notice fur-thermore that every representation ( ρ, V ) of G n extends naturally to a representation( ρ, V ) of Γ by setting ρ ( γ ) := ρ ( p n ( γ )) , γ ∈ Γ . For each finite set Z of words in the alphabet { , . . . , m } and each finite-dimensionalrepresentation ( ρ, V ) of Γ, we define the operator Z ( ρ ) : V → V by Z ( ρ ) = X a ∈ Z ρ ( γ a ) , where for a = a · · · a N we write γ a = γ a ◦ · · · ◦ γ a N . Recall from (19) that W N denotesthe set of reduced words of length N in the alphabet { , . . . , m } which correspondsone-to-one to the set elements of Γ of reduced word length N via the map a γ a . Moreover, we define the operator norm of a linear operator A : V → V by k A k End( V ) = max v ∈ V, k v k V =1 k Av k V . With all this in place we can now state the main result of this section.
Proposition 5.1 (Exponential decay norm estimate for W N ( ρ )) . Assume that thegraphs G n form a family of two-sided ε -expanders. Then for every n and for everynon-trivial, irreducible representation ( ρ, V ) of G n , and for every positive integer N we have k W N ( ρ ) k End( V ) ≤ Ce − cN | W N | , where both C > and c > are positive constants depending only on ε and the number m of Schottky generators.Proof. The group G n acts by linear automorphisms on L ( G n ) through the (left) reg-ular representation R n ( g ) ϕ ( x ) = ϕ ( g − x ) . for all g ∈ G n , x ∈ G n and ϕ ∈ L ( G n ) . This gives a unitary representation R n : G n → U( L ( G n )) . The regular representation can we decomposed into the irreducible repre-sentations of G n , R n = M ( ρ,V ) ∈ b G n dim( ρ ) ρ. This allows us to decompose the Markov operator defined in (40) accordingly as(41) M n = M ρ ∈ b G n dim( ρ ) M ρ = id ⊕ M ρ ∈ b G n dim( ρ ) M ρ where for each ρ , M ρ is the operator M ρ : L ( V ) → L ( V ) , M ρ = 1 | S | X s ∈ S ρ ( s ) . In light of (41), we deduce that the graphs G n form a family of two-sided ε -expandersif and only if there exists ε > n and all non-trivial representations ρ ∈ b G n , the eigenvalues of M ρ , denoted by µ ( ρ ) , . . . , µ d ( ρ ), satisfy the bound(42) max ≤ k ≤ d | µ k ( ρ ) | ≤ − ε. For the remainder of this proof we fix n and a non-trivial irreducible representation( ρ, V ) of G n , and we denote by λ , . . . , λ d the eigenvalues of the operator W ( ρ ).Noticing that W ( ρ ) = 2 mM ρ , we deduce from (42) that the eigenvalues of W ( ρ )satisfy(43) max ≤ k ≤ d | λ k | ≤ m (1 − ε ) , which we will use later in the proof.Now we define W ( ρ ) = I V , where I V denotes the identity operator on V , and weconsider the following formal power series with coefficients in the endomorphism ringEnd( V ) of V :(44) G ρ ( z ) := ∞ X N =0 W N ( ρ ) z N . An elementary argument shows that | W N | ≍ (2 m − N , where the implied constants are independent of N . Hence, from the the trivial estimate k W N ( ρ ) k End( V ) ≤ | W N | , we deduce that (44) is absolutely convergent for | z | < m − with respect to the operatornorm k · k End( V ) .The next step is to derive a closed-form expression for G ρ ( z ). To that effect, note thatwe have the elementary but essential recursion formulas (45) W ( ρ ) = W ( ρ ) + 2 m · I V and for all N ≥ W ( ρ ) W N ( ρ ) = W N +1 ( ρ ) + (2 m − W N − ( ρ ) . This was used by Ihara [21] in his derivation of what came to be known as the Ihara zeta function
We can now multiply G ρ ( z ) by zW ( ρ ) on the right and apply the recursion formulas(45) and (46) to obtain zW ( ρ ) G ρ ( z ) = W ( ρ ) z + W ( ρ ) z + ∞ X N =2 W ( ρ ) W N ( ρ ) z N +1 = W ( ρ ) z + ( W ( ρ ) + 2 m · I V ) z + ∞ X N =2 W N +1 ( ρ ) z N +1 + (2 m − ∞ X N =2 W N − ( ρ ) z N +1 . Inserting ∞ X N =2 W N +1 ( ρ ) z N +1 = G ρ ( z ) − W ( ρ ) z − W ( ρ ) z − I V and ∞ X N =2 W N − ( ρ ) z N +1 = z ( G ρ ( z ) − I V )into the above equation yields zW ( ρ ) G ρ ( z ) = ( z − I V + G ρ ( z ) (cid:0) m − z (cid:1) . Rearranging gives the closed-form expression G ρ ( z ) = (1 − z ) (cid:0) I V − W ( ρ ) z + (2 m − z (cid:1) − . Using this formula, we can now derive an explicit formula for W N ( ρ ). To do so,observe that W ( ρ ) is a self-adjoint operator by virtue of the fact that ρ is a unitaryrepresentation and S is symmetric, i.e., S − = S . Hence, we can find a basis of V withrespect to which W ( ρ ) acts diagonally, i.e., we can write W ( ρ ) = diag( λ , . . . , λ d )where λ , . . . , λ d are the eigenvalues of W ( ρ ). For each k ∈ { , . . . , d } , let ω ± k be thepair of complex numbers satisfying1 − λ k z + (2 m − z = (1 − ω + k z )(1 − ω − k z ) . With respect to the diagonalizing basis, we then have the expression G ρ ( z ) = diag (cid:18) − z (1 − ω + k z )(1 − ω − k z ) (cid:19) dk =1 . Expanding each diagonal entry as a power series in z gives1 − z (1 − ω + k z )(1 − ω − k z ) = ∞ X N =0 ( ξ k,N − ξ k,N − ) z N , where ξ k, − = ξ k, − = 0 and(47) ξ k,N = N X l =0 (cid:0) ω + k (cid:1) l (cid:0) ω − k (cid:1) N − l for all N ≥
0. This gives W N ( ρ ) = diag ( ξ k,N − ξ k,N − ) dk =1 . It follows that(48) k W N ( ρ ) k End( V ) = max ≤ k ≤ d | ξ k,N − ξ k,N − | . A direct computation shows that(49) ω ± k = λ k ± q λ j − m − . Suppose that | λ k | ≤ √ m −
1. In this case it follows from (49) that | ω ± k | = √ m − . Now suppose | λ j | > √ m −
1. In this case we recall from (43) that | λ j | ≤ m (1 − ε ) , which we use to deduce the bound | ω ± k | ≤ | λ k | + p (2 m ) − m − ≤ (2 m − (cid:18) − m m − ε (cid:19) . Therefore, setting ε ′ = min (cid:26) m m − ε, − √ m − (cid:27) > , we obtain(50) max ≤ k ≤ d | ω ± k | ≤ (2 m − − ε ′ ) < (2 m − e − ε ′ . Using (47) and (50), we obtainmax ≤ k ≤ d | ξ k,N | ≤ N (cid:18) max ≤ k ≤ d | ω ± k | (cid:19) N ≤ N e − ε ′ N (2 m − N , Hence, returning to (48) and recalling that | W N | ≍ (2 m − N , we obtain k W N ( ρ ) k End( V ) ≤ max ≤ k ≤ d | ξ k,N | + max ≤ k ≤ d | ξ k,N − |≪ N e − ε ′ N (2 m − N ≪ e − ε ′′ N | W N | , where ε ′′ = ε ′ /
2, say. This completes the proof of Proposition 5.1. (cid:3)
For every positive integer N and for every i, j ∈ { , . . . , m } consider the followingsubset of reduced words of length N : A i,jN = { a = a · · · a N ∈ W N : a = i, a N = j } . It turns out that in our application it is more useful to have a bound on the operatornorm of A i,jN ( ρ ) rather than W N ( ρ ). Corollary 5.2 (Norm estimate for A i,jN ( ρ )) . Assume that the graphs G n form a familyof two-sided ε -expanders. Then for all n , all non-trivial irreducible representations ( ρ, V ) of G n , all N , and all i, j ∈ { , . . . , m } , we have the bound k A i,jN ( ρ ) k End( V ) ≤ Ce − cN | W N | where C > and c > depend only on ε and m .Proof. We define the subset of reduced words B iN = { a = a · · · a N ∈ W N : a = i } . Notice that we have the identity B iN ( ρ ) = ρ ( γ i ) (cid:0) W N − ( ρ ) − B i + mN − ( ρ ) (cid:1) . Applying the triangle inequality and using the fact that ρ is unitary gives(51) k B iN ( ρ ) k End( V ) ≤ k W N − ( ρ ) k End( V ) + k B i + mN − ( ρ ) k End( V ) . Hence, setting b N ( ρ ) := max ≤ i ≤ m k B iN ( ρ ) k End( V ) , we obtain b N ( ρ ) ≤ k W N − ( ρ ) k End( V ) + b N − ( ρ ) . Iterating this inequality gives(52) b N ( ρ ) ≤ N − X k =0 k W k ( ρ ) k End( V ) . By Proposition 5.1, we have k W k ( ρ ) k End( V ) ≪ e − ck | W k | ≪ e − ck (2 m − k . By inserting this bound into (52), using the geometric series summation and the factthat | W N | ≍ (2 m − N , we obtain(53) b N ( ρ ) ≪ N − X k =0 e − ck (2 m − k ≪ e − cN (2 m − N ≪ e − cN | W N | . Notice that we also have the relation(54) A i,jN ( ρ ) = (cid:16) B iN − ( ρ ) − A i,j + mN ( ρ ) (cid:17) ρ ( γ j ) . We can use arguments similar to the ones above. Setting a N ( ρ ) := max ≤ i,j ≤ m k A i,jN ( ρ ) k End( V ) , we derive from (54) a N ( ρ ) ≤ k B iN − ( ρ ) k End( V ) + a N − ( ρ ) ≤ b N − ( ρ ) + a N − ( ρ ) . By iterating this bound, we get a N ( ρ ) ≤ N − X k =0 b k ( ρ ) , which we can combine with the bound in (53) to obtain a N ( ρ ) ≪ e − cN | W N | . This concludes the proof of Corollary 5.2. (cid:3)
Decoupling estimate.
In this subsection we strengthen Proposition 5.1. Recallthat for every function g : Λ → V we set k g k ∞ , Λ := max x ∈ Λ k g ( x ) k V . Moreover, we introduce the semi-norm[ g ] Λ := max ≤ j ≤ m max x,y ∈ Λ ∩ I j (cid:13)(cid:13)(cid:13)(cid:13) g ( x ) − g ( y ) x − y (cid:13)(cid:13)(cid:13)(cid:13) V , where I j = D j ∩ R are the real intervals spanned by the Schottky disks D , . . . , D m of Γ. With this in place we can now state the next result. Proposition 5.3 (Decoupling estimate) . Assume that the graphs G n form a family oftwo-sided ε -expanders. Then for every n , for every non-trivial, irreducible represen-tation ( ρ, V ) of G n , for every function g : Λ → V , for every j ∈ { , . . . , m } , and forevery x ∈ Λ ∩ I j , we have the bound | W jN | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≤ Ce − cN ( k g k ∞ , Λ + [ g ] Λ ) , where the constants c > and C > depend only on ε and the Schottky data of Γ .Remark . Note that by specializing Proposition 5.3 to the constant 1 function g = ,we recover the purely combinatorial statement in Proposition 5.1. Proof.
For each pair j, l ∈ { , . . . , m } we define the set Z l,jN = { a = a · · · a N ∈ W N : a = l + m and a N = j } , which corresponds precisely to the set of reduced words a ∈ W N for which γ a mapsthe disk D j to the interior of D l . We can write Z l,jN ( ρ ) = X ≤ j ′ ≤ mj ′ = j A l + m,j ′ N ( ρ )where A i,jN = { a = a · · · a N ∈ W N : a = i, a N = j } . Applying Corollary 5.2 gives(55) k Z l,jN ( ρ ) k End( V ) ≤ X ≤ j ′ ≤ mj ′ = j k A l + m,j ′ N ( ρ ) k End( V ) ≪ e − cN | W N | , where both c > ε and the Schottky dataof Γ. Note that the sets Z l,jN and W N are of comparable size, that is, we have | Z l,jN | ≍ | W N | with some absolute implied constants. Combining this with (55) yields the estimate(56) k Z l,jN ( ρ ) k End( V ) ≪ e − cN | Z l,jN | , which will be used later in the proof.We will use a decoupling argument similar to the one contained in [7, § N = N + N where N = ⌊ N/ ⌋ and N = ⌈ N/ ⌉ and notice that for every j ∈ { , . . . , m } and every word a ∈ W jN there exists a unique l ∈ { , . . . , m } and unique words a ∈ W lN and a ∈ Z l,jN such that a = a a . (Here, a a stands for the concatenationof the words a and a .) With this in mind, we write X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) = m X l =1 X a ∈ W lN , a ∈ Z l,jN ρ ( γ a a ) − g ( γ a a ( x ))= m X l =1 X a ∈ W lN , a ∈ Z l,jN ρ ( γ a a ) − g ( γ a a ( x )) | {z } = S l . For each g ( γ a a ( x )) appearing in the sum above we subtract and add the term1 | Z l,jN | X b ∈ Z l,jN g ( γ a γ b ( x )) . We obtain a decomposition for each S l into a sum of two parts: S l = X a ∈ W lN , a ∈ Z l,jN ρ ( γ a a ) − g ( γ a a ( x )) − | Z l,jN | X b ∈ Z l,jN g ( γ a γ b ( x )) | {z } = S l + X a ∈ W lN , a ∈ Z l,jN ρ ( γ a a ) − | Z l,jN | X b ∈ Z l,jN g ( γ a γ b ( x )) | {z } = S l . We will now consider the two parts S l and S l separately. Notice that we can rewrite S l as S l = 1 | Z l,jN | X a , b ∈ Z l,jN X a ∈ W lN ρ ( γ a a ) − ( g ( γ a a ( x )) − g ( γ a b ( x ))) . We use the triangle-inequality and the unitarity of ρ to estimate k S l k V ≤ | Z l,jN | X a , b ∈ Z l,jN X a ∈ W lN k g ( γ a a ( x )) − g ( γ a b ( x )) k V ≤ | Z l,jN | X a , b ∈ Z l,jN X a ∈ W lN [ g ] Λ ( γ a ( γ a ( x )) − γ a ( γ b ( x ))) . By the definition of the set Z j,lN , the following holds true: for all a , b ∈ Z l,jN and eachpoint x ∈ Λ ∩ I j , the images γ a ( x ) and γ b ( x ) are contained in D l . Hence, we can useLemma 4.6 to estimate the inner sum as follows: X a ∈ W lN ( γ a ( γ a ( x )) − γ a ( γ b ( x ))) ≪ e N P (1) . Here, P ( · ) is the pressure defined in (27) and the implied constant depends only onthe Schottky data of Γ. Inserting this into the above bound for k S l k V gives(57) k S l k V ≪ | Z l,jN | e N P (1) [ g ] Λ . Let us now focus on S l . Since ρ is a representation, we have ρ ( γ a a ) = ρ ( γ a ) ρ ( γ a ).Thus we can interchange the order of summation in S l to obtain S l = X a ∈ W lN , a ∈ Z l,jN ρ ( γ a a ) − | Z l,jN | X b ∈ Z l,jN g ( γ a γ b ( x )) = 1 | Z l,jN | X a ∈ W lN , b ∈ Z l,jN X a ∈ Z l,jN ρ ( γ a ) − ρ ( γ a ) − g ( γ a γ b ( x )) . Since ρ is unitary, we have ρ ( γ ) − = ρ ( γ ) ∗ for all γ ∈ Γ, where A ∗ denotes the adjointof A . Thus, we have X a ∈ Z l,jN ρ ( γ a ) − = X a ∈ Z l,jN ρ ( γ a ) ∗ = Z l,jN ( ρ ) ∗ , which we insert above to obtain S l = 1 | Z l,jN | X a ∈ W lN , b ∈ Z l,jN Z l,jN ( ρ ) ∗ ρ ( γ a ) − g ( γ a γ b ( x )) . Applying the triangle inequality and using the fact that the operator norm satisfies k A ∗ k End( V ) = k A k End( V ) gives k S l k V ≤ | Z l,jN | X a ∈ W lN , b ∈ Z l,jN k Z l,jN ( ρ ) ∗ ρ ( γ a ) − g ( γ a γ b ( x )) k V ≤ k Z l,jN ( ρ ) k End( V ) | Z l,jN | X a ∈ W lN , b ∈ Z l,jN k g ( γ a γ b ( x )) k V Invoking the bound in (56) yields(58) k S l k V ≪ e − cN X a ∈ W lN , b ∈ Z l,jN k g ( γ a γ b ( x )) k V ≪ e − cN | W jN |k g k ∞ , Λ . Hence, putting together (57) and (58) gives k S l k V ≤ k S l k V + k S l k V ≪ e − cN | W jN |k g k ∞ , Λ + | Z l,jN | e P (1) N [ g ] Λ . Recalling that N = ⌊ N/ ⌋ , N = ⌈ N/ ⌉ and P (1) <
0, and noticing the trivial bound | Z l,jN | < | W jN | , we obtain k S l k V ≪ e − c N | W jN | ( k g k ∞ , Λ + [ g ] Λ ) , with c = min (cid:26) c , − P (1)2 (cid:27) > . Consequently, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≤ m X l =0 k S l k V ≪ e − c N | W jN | ( k g k ∞ , Λ + [ g ] Λ )which concludes the proof. (cid:3) Norm Estimate for L Nδ,ρ
We will keep all the notations of the previous sections. The goal of this section is toprove norm estimates for L Nδ,ρ which are exponentially decaying as N → ∞ , the decayrate being uniform among all non-trivial, irreducible representations ρ of the coveringgroups G n = Γ / Γ n , provided that the graphs G n form a family of two-sided expanders. Proposition 6.1 (Norm estimate for L Nδ,ρ ) . Assume that the graphs G n form a family oftwo-sided ε -expanders. Then there exists a positive integer N such that for every n , forevery non-trivial, irreducible representation ( ρ, V ) of G n , and for every f ∈ H ( D, V ) ,we have kL N δ,ρ f k ≤ k f k . Here, N depends only on ε and the Schottky data of Γ . Moreover, for every positiveinteger N , we have kL Nδ,ρ f k ≤ Ce − cN k f k and kL Nδ,ρ f k ∞ , Λ ≤ Ce − cN k f k where the constants c > and C > depend solely on ε and the Schottky data of Γ . Remark . Here and always, k · k denotes the L -norm given in (29) while k f k ∞ , Λ =sup x ∈ Λ k f ( x ) k V is the maximum norm of f on the limit set.The proof of Proposition 6.1 will occupy this entire section and will be divided intoseveral lemmas. We will initially work with the normalized operator(59) e L δ,ρ = ϕ − L δ,ρ ϕ, where ϕ = ϕ δ is the 1-eigenfunction of L δ furnished by the Ruelle–Perron–Frobeniustheorem (Proposition 4.1). The operator e L δ,ρ is “normalized” in the sense that(60) e L δ, id ( ) = , where id is the trivial one-dimensional representation and is the constant 1 functionon Λ . Since ϕ is positive on the limit set Λ, (59) defines an operator e L δ,ρ : C (Λ , V ) → C (Λ , V )on the set of functions C (Λ , V ) = { g : Λ → V } . Using the formula (20) for the iteratesof L s,ρ , we obtain for all g ∈ C (Λ , V ), all j ∈ { , . . . , m } and all x ∈ Λ ∩ I j ,(61) e L Nδ,ρ g ( x ) = ( ϕ L Nδ,ρ ϕ − ) g ( x ) = X a ∈ W jN w a ( x ) ρ ( γ a ) − g ( γ a ( x )) , where(62) w a ( x ) = ϕ ( γ a ( x )) ϕ ( x ) γ ′ a ( x ) δ are positive weights satisfying X a ∈ W jN w a ( x ) = 1 . In particular, for every fixed point x ∈ Λ ∩ I j , the vector e L Nδ,ρ g ( x ) (viewed as a vectorin V ) is a convex linear combination of the vectors ρ ( γ a ) − g ( γ a ( x )) with a ∈ W jN . Wewill use the following elementary statement, which is a quantitative expression of thefact that the unit sphere in V is strictly convex: Lemma 6.3 (strict convexity) . Let V be a complex vector space endowed with thehermitian inner product h· , ·i V and induced norm k v k V = p h v, v i V , and let M > .Consider the convex linear combination in V , (63) w = k X i =1 a i v i , where all the vectors v , . . . , v k satisfy k v i k ≤ M and a , . . . , a k are positive reals sum-ming up to . Then, if k w k V ≥ M (1 − ν ) for some < ν < , we have (64) k w k V ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k k X i =1 v i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V + 2 M √ ν min ≤ j ≤ k a j . Remark . In the above statement, note that ν = 0 forces w = v = · · · = v k , whichis to say that the sphere { v ∈ V : k v k V = M } is strictly convex. Proof.
Squaring the norm of (63), we obtain(65) k w k V = X ≤ i,j ≤ k a i a j Re h v i , v j i V . Using the relation Re h v i , v j i V = 12 ( k v i k V + k v j k − k v i − v j k V ) . and the assumption that k v i k V ≤ M , we obtain(66) h v i , v j i V ≤ M − k v i − v j k V . Inserting (66) into (65) and using the fact that a , . . . , a k are positive reals that sumup to 1, gives k w k V ≤ M − X ≤ i,j ≤ k a i a j k v i − v j k V , which we can rearrange to give(67) X ≤ i,j ≤ k a i a j k v i − v j k V ≤ M − k w k ) . Using the assumption that k w k V ≥ M (1 − ν ) gives X ≤ i,j ≤ k a i a j k v i − v j k V ≤ M (2 ν − ν ) < M ν. Thus, for all 1 ≤ i, j ≤ k we obtain the bound k v i − v j k V < M √ ν √ a i a j ≤ M √ ν min ≤ j ≤ k a j . Consequently, we have(68) k w − v i k V ≤ k X j =1 a j k v i − v j k V < M √ ν min ≤ j ≤ k a j . The statement of Lemma 6.3 now follows from (68) and w = 1 k k X i =1 v i + 1 k k X i =1 ( w − v i )after applying the triangle inequality. (cid:3) Recall that for every function g : Λ → V we define k g k ∞ , Λ := max x ∈ Λ k g ( x ) k V . and [ g ] Λ := max ≤ j ≤ m max x,y ∈ Λ ∩ I j (cid:13)(cid:13)(cid:13)(cid:13) g ( x ) − g ( y ) x − y (cid:13)(cid:13)(cid:13)(cid:13) V . For any c > F c,V = { g : Λ → V : [ g ] Λ ≤ c k g k ∞ , Λ } . We can now state the next result.
Lemma 6.5.
Assume the graphs G n form a family of two-sided ε -expanders and fixa constant c > . Then there are constants ν = ν ( ε, c ) > and N = N ( ε, c ) ∈ N depending only on ε , c and the Schottky data of Γ such that for every n , every non-trivial, irreducible representation ( ρ, V ) of G n , and every g ∈ F c,V , we have k e L Nδ,ρ g k ∞ , Λ ≤ (1 − ν ) k g k ∞ , Λ . Proof.
Fix n , a non-trivial representation of ( ρ, V ) of G n , and a function g ∈ F c,V .Recall from Proposition 5.3 that there are positive constants c , c depending solelyon ε and the Schottky data of Γ such that for all positive integers N we have(69) 1 | W jN | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≤ c e − c N ( k g k ∞ , Λ + [ g ] Λ ) . Using the assumption that g ∈ F c,V , this implies(70) 1 | W jN | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≤ c e − c N k g k ∞ , Λ , where we have set c = c (1 + c ). For the remainder of this proof, let N = (cid:24) log(2 c + 1) c (cid:25) , so that(71) 1 | W jN | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≤ k g k ∞ , Λ . Set c = min ≤ j ≤ m min a ∈ W jN min x ∈ Λ ∩ I j w a ( x ) ! − > , where the w a ( x )’s are the positive weights in (62) and set(72) ν = 19(1 + c ) . Notice that both N and ν depend only on ε , c and the Schottky data of Γ. We wantto show that(73) k e L Nδ,ρ g k ∞ , Λ ≤ (1 − ν ) k g k ∞ , Λ . Let x ∈ Λ be the point which maximizes the function k e L Nδ,ρ g ( x ) k V , i.e., choose j and x ∈ Λ ∩ I j such that k e L Nδ,ρ g k ∞ , Λ = k e L Nδ,ρ g ( x ) k V . Suppose by contradiction that we have(74) k e L Nδ,ρ g ( x ) k V > (1 − ν ) k g k ∞ , Λ . Recall from (61) that e L Nδ,ρ g ( x ) is the convex linear combination of the vectors v a = ρ ( γ a ) − g ( γ a ( x )) ∈ V , e L Nδ,ρ g ( x ) = X a ∈ W jN w a ( x ) ρ ( γ a ) − g ( γ a ( x )) . Notice that since ρ is unitary, we have k v a k ≤ k g k ∞ , Λ . Thus we can apply Lemma 6.3to obtain k e L Nδ,ρ g ( x ) k V ≤ | W jN | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN ρ ( γ a ) − g ( γ a ( x )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V + c √ ν k g k ∞ , Λ . By the choice of N and ν this gives k e L Nδ,ρ g ( x ) k V ≤ (cid:18)
12 + c √ ν (cid:19) k g k ∞ , Λ < k g k ∞ , Λ < (1 − ν ) k g k ∞ , Λ , contradicting (74). This concludes the proof. (cid:3) Lemma 6.6.
There exists a constant
C > such that for every finite-dimensional,unitary representation ( ρ, V ) of Γ , every function g : Λ → V , and every positive integer N , we have [ e L Nδ,ρ g ] ≤ C ( k g k ∞ , Λ + [ g ] Λ ) . Proof.
Recall from (61) that for all j ∈ { , . . . , m } and all x ∈ Λ ∩ I j , e L Nδ,ρ g ( x ) = X a ∈ W jN w a ( x ) ρ ( γ a ) − g ( γ a ( x )) . Now let x, y ∈ Λ ∩ I j and write e L Nδ,ρ g ( x ) − e L Nδ,ρ g ( y ) = X a ∈ W jN ρ ( γ a ) − ( w a ( x ) g ( γ a ( x )) − w a ( y ) g ( γ a ( y ))) . By the unitarity of ρ , we get(75) k e L Nδ,ρ g ( x ) − e L Nδ,ρ g ( y ) k V ≤ X a ∈ W jN k w a ( x ) g ( γ a ( x )) − w a ( y ) g ( γ a ( y )) k V . Notice also that k w a ( x ) g ( γ a ( x )) − w a ( y ) g ( γ a ( y )) k V ≤ k w a k ∞ ,I j k g ( γ a ( x )) − g ( γ a ( y )) k V + k g k ∞ , Λ k w a ( x ) − w a ( y ) k≤ k w a k ∞ ,I j [ g ] Λ | x − y | + k g k ∞ , Λ k w a ( x ) − w a ( y ) k Recall from (62) that the weights w a are given by(76) w a ( x ) = ϕ ( γ a ( x )) ϕ ( x ) γ ′ a ( x ) δ , where ϕ = ϕ δ ∈ C ( I ) is the 1-eigenfunction of L δ provided by Proposition 4.1. Itimmediately follows that(77) k w a k ∞ ,I j ≪ k γ ′ a k δ ∞ ,I j . Since the function x w a ( x ) belongs to C ( I ), we can apply the mean value theoremto obtain(78) k w a ( x ) − w a ( y ) k ≤ k w ′ a k ∞ ,I j | x − y | . Using (76), a direct computation using the chain and product rules gives w ′ a ( x ) = δ ϕ ( γ a ( x )) ϕ ( x ) γ ′′ a ( x ) γ ′ a ( x ) δ − + ϕ ′ ( γ a ( x )) ϕ ( x ) γ ′ a ( x ) δ +1 − ϕ ( γ a ( x )) ϕ ′ ( x ) ϕ ( x ) γ ′ a ( x ) δ . We deduce the bound k w ′ a k ∞ ,I j ≪ k γ ′′ a · ( γ ′ a ) δ − k ∞ ,I j + k γ ′ a k δ +1 ∞ ,I j + k γ ′ a k δ ∞ ,I j = (cid:13)(cid:13)(cid:13)(cid:13) γ ′′ a γ ′ a (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,I j · k γ ′ a k δ ∞ ,I j + k γ ′ a k δ +1 ∞ ,I j + k γ ′ a k δ ∞ ,I j By the bounded distortion estimate (22), we have (cid:13)(cid:13)(cid:13)(cid:13) γ ′′ a γ ′ a (cid:13)(cid:13)(cid:13)(cid:13) ∞ ,I j ≪ , thus we obtain(79) k w ′ a k ∞ ,I j ≪ k γ ′ a k δ ∞ ,I j with some implied constant depending only on the Schottky data of Γ. Putting together(77), (78) and (79), we obtain k w a ( x ) g ( γ a ( x )) − w a ( y ) g ( γ a ( y )) k V ≪ k γ ′ a k δ ∞ ,I j ( k g k ∞ , Λ + [ g ] Λ ) | y − x | Inserting this into (75) and using Lemma 4.2, we get for all x, y ∈ I j the bound k e L Nδ,ρ g ( x ) − e L Nδ,ρ g ( y ) k V | x − y | ≪ ( k g k ∞ , Λ + [ g ] Λ ) X a ∈ W jN k γ ′ a k δ ∞ ,I j ≪ k g k ∞ , Λ + [ g ] Λ , with some implied constant depending solely on the Schottky data of Γ. Since x and y were arbitrary points in I j , the statement follows. (cid:3) The next result is a strengthening of Lemma 6.5.
Lemma 6.7.
With the same assumptions and notations as in Proposition 6.1 (andLemma 6.5), let η > and c > be fixed constants. Then there exists a positive integer N depending only on ε , η , c and the Schottky data of Γ such that for all g ∈ F c,V wehave k e L N δ,ρ g k ∞ , Λ ≤ η k g k ∞ , Λ Proof.
By assumption, the graphs G n form a family of two-sided ε -expanders. Wefix an index n , a non-trivial, irreducible representation ( ρ, V ) of G n , and a function g ∈ F c,V . Suppose by contradiction that we have(80) k e L Nδ,ρ g k ∞ , Λ > η k g k ∞ , Λ for all positive integers N . The goal is to show that (80) is false for some N dependingonly on η and c . In what follows, c κ ( κ ∈ N ) denote positive constants which are only allowed to depend on ε , η , c , and the Schottky data of Γ. Applying Lemma 6.6 andusing the assumption that g ∈ F c,V yields(81) [ e L Nδ,ρ g ] ≤ c k g k ∞ , Λ for all N . Combining (80) and (81), we obtain for all N ,(82) [ e L Nδ,ρ g ] ≤ c k e L Nδ,ρ g k ∞ , Λ with c = c η − . In other words, the assumption in (80) forces e L Nδ,ρ g ∈ F c for all positive integers N . We now set c = max { c , c } . Then we have(83) g, e L δ,ρ g, e L δ,ρ g, e L δ,ρ g, · · · ∈ F c . By Lemma 6.5 there exist ν = ν ( ε, c ) > k = k ( ε, c ) ∈ N depending only on ε, c ,and the Schottky data of Γ such that for all h ∈ F c , k e L kδ,ρ h k ∞ , Λ ≤ (1 − ν ) k h k ∞ , Λ . In particular, for all integers m ≥
2, we have k e L mkδ,ρ g k ∞ , Λ = k e L kδ,ρ ( e L ( m − kδ,ρ g | {z } ∈F c ) k ∞ , Λ ≤ (1 − ν ) k e L ( m − kδ,ρ g k ∞ , Λ . We can now iterate this to give k e L mkδ,ρ g k ∞ , Λ ≤ (1 − ν ) m k g k ∞ , Λ ≤ e − νm k g k ∞ , Λ . But this contradicts (80) for m > ν − log η − . Hence, Lemma 6.7 follows with N = (cid:6) kν − log η − (cid:7) , which depends only on ε, c , and the Schottky data of Γ . (cid:3) Lemma 6.8.
With the same assumptions and notations as in Proposition 6.1, let η > and c > be fixed constants. Then there exists a positive integer N = N ( ε, η, c ) such that for all n , all non-trivial, irreducible representations ( ρ, V ) , and all functions f ∈ H ( D, V ) with f | Λ ∈ F c,V , we have kL Nδ,ρ f k ≤ η k f k . Here, f | Λ denotes the restriction of f to the limit set Λ .Proof. Fix a function f ∈ H ( D, V ) with f | Λ ∈ F c,V and notice that on Λ, we have(84) L Nδ,ρ f = ϕ e L Nδ,ρ e f where(85) e f = f | Λ ϕ . One easily verifies that e f ∈ F c for some c > c and ϕ . Hence,applying Lemma 6.7, we obtain for any η > N ′ depending onlyon ε , η , c and the Schottky data of Γ, such that(86) k e L N ′ δ,ρ e f k ∞ , Λ ≤ η k e f k ∞ , Λ . This bound will be used further below. In what follows, the constants c κ with κ ∈ N will denote positive constants independent of n and ρ . By (84) we have(87) kL Nδ,ρ f k ∞ , Λ ≤ c k e L Nδ,ρ e f k ∞ , Λ , where c = max Λ | ϕ | and by (85) we have(88) k e f k ∞ , Λ ≤ c k f k ∞ , Λ where c = (min Λ | ϕ | ) − . By combining (86), (87) and (88), we get(89) kL N ′ δ,ρ f k ∞ , Λ ≤ c c η k f k ∞ , Λ . Recall from Lemma 4.7 that there exists positive constants c , c , c such that for allpositive integers N the following estimates hold true:(90) kL Ns,ρ f k ≤ c (cid:0) k f k ∞ , Λ + e − c N k f k (cid:1) and(91) kL Ns,ρ f k ≤ c k f k . Furthermore, recall from Lemma 4.5 that there exists c > k f k ∞ , Λ ≤ c k f k . Now we choose N ′′ = (cid:6) c − log(1 + η − ) (cid:7) so that(93) e − c N ′′ < η and we set N = N ′ + N ′′ . Notice that N depends only on ε , η , c and the Shottkydata of Γ. The quantity η will be specified below in terms of η . Applying the aboveestimates, we obtain kL N s,ρ f k = kL N ′′ δ,ρ ( L N ′ δ,ρ f ) k ≤ c (cid:16) kL N ′ δ,ρ f k ∞ , Λ + e − c N ′′ kL N ′ s,ρ f k (cid:17) by (90) ≤ c (cid:16) kL N ′ δ,ρ f k ∞ , Λ + η kL N ′ s,ρ f k (cid:17) by (93) ≤ c (cid:0) c c η k f k ∞ , Λ + c η k f k (cid:1) by (89) ≤ c c (cid:0) c c + 1 (cid:1) η k f k by (92).We can now choose η > c c (cid:0) c c + 1 (cid:1) η = η , giving kL N s,ρ f k ≤ η k f k . This completes the proof of Lemma 6.8. (cid:3)
Lemma 6.9.
There exists
C > depending only on the Schottky data of Γ such thatfor all f ∈ H ( D, V ) , we have [ f ] Λ ≤ C k f k . Proof.
Let d = dim( V ) be the dimension of V . Fix an orthonormal basis e , . . . , e d of V and decompose f as f = f e + · · · + f d e k where f , . . . , f d ∈ H ( D ), so that k f k = k f k + · · · + k f d k . Notice that(94) [ f ] ≤ [ f ] + · · · + [ f d ] , so it suffices to show that(95) [ f k ] Λ ≪ k f k k for all k ∈ { , . . . , d } , the implied constant depending solely on the Schottky data of Γ.To that effect, fix k ∈ { , . . . , d } , j ∈ { , . . . , m } and two points x, y ∈ Λ ∩ I j . Thanksto the mean value theorem, there exists a point x ′ in the line segment connecting x and y such that f k ( y ) − f k ( x ) y − x = f ′ k ( x ′ ) . Since the line segment line segment connecting x and y is bounded away from ∂D j uniformly in x, y , there exists c > x ′ , ∂D ) ≥ c. Hence, Lemma 4.3 implies that (cid:12)(cid:12)(cid:12)(cid:12) f k ( y ) − f k ( x ) y − x (cid:12)(cid:12)(cid:12)(cid:12) ≪ k f k k , where the implied constant depends only on the Schottky data of Γ. Since the points x, y were taken arbitrarily from Λ ∩ I j , the bound in (95) follows, completing the proofof Lemma 6.9. (cid:3) We are now ready for the proof of Proposition 6.1.
Proof of Proposition 6.1.
Assume that the graphs G n form a family of two-sided ε -expanders. Let us recall what we want to show. For each n , for each non-trivial,irreducible representation ( ρ, V ) of G n , and for each functions f ∈ H ( D, V ), we wantto show that there exists a positive integer N depending only on ε and the Schottkydata of Γ such that(96) kL N s,ρ f k ≤ k f k . Moreover, we want to show that there exist positive constants c , C and C ′ (again,depending only on ε and the Schottky data of Γ) such that(97) kL Nδ,ρ f k ≤ Ce − cN k f k and(98) kL Nδ,ρ f k ∞ , Λ ≤ C ′ e − cN k f k . In what follows, the constants c κ , κ ∈ N are uniform in n, ρ and f . We claim that itsuffices to prove (96). Indeed, by Lemma 4.5, there exists c > kL Nδ,ρ f k ∞ , Λ ≤ c kL Nδ,ρ f k Hence, the bound in (98) follows from (97) and (99). Recall also from Lemma 4.7 thatthere exists c > M ,(100) kL Ms,ρ f k ≤ c k f k . To see how (97) follows from (96), notice that we can write N = kN + m where k and0 ≤ m < N are non-negative integers. Then we have kL Ns,ρ f k = kL ms,ρ ( L kN s,ρ f ) k ≤ c kL kN s,ρ f k ≤ c ( 12 ) k k f k ≤ Ce − cN k f k , which proves (97) with C = 2 c and c = log 2 N .The rest of this proof is dedicated to prove (96). Recall from Lemma 4.7 that thereexist c > c > kL Ns,ρ f k ≤ c (cid:0) k f k ∞ , Λ + e − c N k f k (cid:1) . We choose N = (cid:24) log(8 c c + 1) c (cid:25) so that 4 c c e − c N < Case 1 : assume that kL N s,ρ f k ≤ c k f k . Case 2 : assume the opposite, so kL N s,ρ f k > c k f k , or equivalently,(102) k f k < c kL N s,ρ f k . We claim that in the second case, the restriction f | Λ of f to the limit set belongs to F c,V for some c > n, ρ, f . To see this, we combine (102) and (101) to get(103) k f k < c kL N s,ρ f k ≤ c c ( k f k ∞ , Λ + e − c N k f k ) . By the choice of N this implies k f k < c c k f k ∞ , Λ + 12 k f k , which can be rearranged to give(104) k f k < c c k f k ∞ , Λ . By Lemma 6.9, there exists c > f ] Λ ≤ c k f k . Combining (105) and (104) gives[ f ] ≤ c c c k f k ∞ , Λ Hence, in Case 2 we have f | Λ ∈ F c with c = p c c c .But now we can apply Lemma 6.8 to obtain a positive integer N depending only on ε and the Schottky data of Γ such that kL N s,ρ f k ≤ c k f k . In conclusion, in either case we havemin {kL N s,ρ f k , kL N s,ρ f k} ≤ c k f k . Set N = N + N and notice that N depends only on ε and the Schottky data of Γ.Using (100) again, this gives kL N s,ρ f k ≤ k f k , completing the proof. (cid:3) Proof of Theorem 1.3 and Theorem 1.4
Proof of Theorem 1.3.
We are now ready to prove the first main result of thispaper, Theorem 1.3. Let Γ be a non-elementary Schottky group and let Γ n be a familyof finite-index, normal subgroups of Γ. We assume that the associated Cayley graphs G n constructed in § ε -expanders. We want to show thatthere exists η = η ( ε ) > n the Selberg zeta function Z Γ n ( s ) hasno non-trivial zeros in the disk | s − δ | < η. Recall that s ∈ C is called a “trivial” zeroof Z Γ n ( s ) if it is a zero of Z Γ ( s ) . In light of the Venkov–Zograf factorization formula (16), Z Γ n ( s ) = Y ρ ∈ b G n Z Γ ( s, ρ ) dim( ρ ) = Z Γ ( s ) Y ρ ∈ b G n Z Γ ( s, ρ ) dim( ρ ) , establishing Theorem 1.3 amounts to proving that for all n and all non-trivial irre-ducible representations ρ ∈ b G n , the twisted Selberg zeta function Z Γ ( s, ρ ) has no zerosin | s − δ | < η .Fix n and ρ ∈ b G n , and assume that Z Γ ( s, ρ ) vanishes at some point s with | s − δ | < η .By the Fredholm determinant identity in Proposition 2.1 and the general theory ofFredholm determinants, the transfer operator L s,ρ defined in (10) must have a non-zero 1-eigenfunction f ∈ H ( D, V ), that is, we have L Ns,ρ f = f for all integers N ≥ . In particular, this implies that(106) kL Ns,ρ f k ≥ k f k for all N ≥ . The strategy of the proof is to show that this gives a contradictionprovided | s − δ | < η , for some sufficiently small η > ε and theSchottky data of Γ. By Proposition 6.1, there exists N ≥ ε andthe Schottky data of Γ such that kL N δ,ρ f k ≤ k f k . Recall from (20) that L N s,ρ f ( z ) = X a ∈ W jN γ ′ a ( z ) s ρ ( γ a ) − f ( γ a ( z )) , z ∈ D j . In what follows, we assume that s ∈ C satisfies | s − δ | <
1, say. Note then that thereexists a constant
C > n and ρ such that for all j ∈ { , . . . , m } , all a ∈ W jN and all z ∈ D j we have | γ ′ a ( z ) s − γ ′ a ( z ) δ | < C | s − δ | . Using this together with Lemma 4.3, we obtain for all j ∈ { , . . . , m } and all z ∈ D j , kL N s,ρ f ( z ) − L N δ,ρ f ( z ) k V ≤ X a ∈ W jN | γ ′ a ( z ) s − γ ′ a ( z ) δ |k f ( γ a ( z )) k V ≤ C ′ | s − δ |k f k , for some constant C ′ > n , ρ and f . It follows that kL N s,ρ f − L N δ,ρ f k ≤ C ′′ | s − δ |k f k for some C ′′ > n , ρ and f . We can now set η = min (cid:8) , C ′′ (cid:9) so that forall s with | s − δ | < η we have kL N s,ρ f − L N δ,ρ f k < k f k . But this implies kL N s,ρ f k < kL N δ,ρ f k + 13 k f k ≤ k f k + 13 k f k < k f k , contradicting (106). The proof of Theorem 1.3 is finished.7.2. Effective equidistribution estimate.
Heuristically, we expect the sums X a ∈ W jN p n ( γ a )= g γ ′ a ( x ) δ to be uniformly distributed among the elements g ∈ G n . For the proof of Theorem1.4, we need an effective equidistribution statement that controls the dependence ofthe error term on n and N : Proposition 7.1 (Effective equidistribution) . Assume the graphs G n form a family oftwo-sided ε -expanders. Then for every n , for every g ∈ G n , for every j ∈ { , . . . , m } ,and for every x ∈ Λ ∩ I j , we have (107) X a ∈ W jN p n ( γ a )= g γ ′ a ( x ) δ = 1 | G n | X a ∈ W jN γ ′ a ( x ) δ + O ( e − cN ) , where both c > and the implied constant in the error term depend solely on ε and theSchottky data of Γ . We will actually use the following result:
Corollary 7.2 (Flattening) . Assume the graphs G n form a family of two-sided ε -expanders. Then there exist constants c > and c > depending only on ε andthe Schottky data of Γ such that the following holds true: for all n , all g ∈ G n , all j ∈ { , . . . , m } , and all N ≥ c log | G n | , we have (108) X a ∈ W jN p n ( γ a )= g k γ ′ a k δ ∞ ,I j ≤ c | G n | . Proof.
The claim follows directly from Proposition 7.1 by taking c = 1 /c , and usingthe uniform hyperbolicity bound in (21) together with the pressure estimate in Lemma4.2. (cid:3) Proof of Proposition 7.1.
Recall that for a group G , we let b G denote the set of itsirreducible representations and we set b G = b G r { id } , where id denotes the trivial,one-dimensional representation of G . Moreover, for a finite-dimensional representation( ρ, V ) of Γ, we let χ ρ ( γ ) = tr V ρ ( γ )be the associated character. We claim that for all j ∈ { , . . . , m } , all x ∈ Λ ∩ I j , and all ( ρ, V ) ∈ b G n , we have thebound(109) X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ = O (dim( ρ ) e − cN ) , where both c > ε and theSchottky data of Γ. Here, we slightly abuse notation to write χ ρ ( gγ − a ) := χ ρ ( gp n ( γ a ) − ) , where p n : Γ → G n is the natural projection map.Before we prove this bound, let us see how it implies (107). Decomposing the charac-teristic function p n ( γ a )= g into irreducible characters as p n ( γ a )= g = 1 | G n | X ρ ∈ b G n dim( ρ ) χ ρ ( gγ − a )= 1 | G n | + 1 | G n | X ρ ∈ b G n dim( ρ ) χ ρ ( gγ − a ) , we can write X a ∈ W jN p n ( γ a )= g γ ′ a ( x ) δ = 1 | G n | X a ∈ W jN γ ′ a ( x ) δ + 1 | G n | X ρ ∈ b G n dim( ρ ) X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ . Thus, assuming the bound in (109), we obtain X a ∈ W jN p n ( γ a )= g γ ′ a ( x ) δ = 1 | G n | X a ∈ W jN γ ′ a ( x ) δ + O | G n | X ρ ∈ b G n dim( ρ ) e − cN = 1 | G n | X a ∈ W jN γ ′ a ( x ) δ + O (cid:0) e − cN (cid:1) , as claimed, where we have used the well-known dimension formula X ρ ∈ b G n dim( ρ ) = | G n | . Now let us prove (109). Fix some ρ ∈ b G n . First, we claim that for all g ∈ G n we havethe relation(110) dim( ρ ) | G n | X h ∈ G n ρ ( h − gh ) = χ ρ ( g ) I V , where I V denotes the identity operator of V. To see this, notice that the operator U g : V → V defined by(111) U g := dim( ρ ) | G n | X h ∈ G n ρ ( h − gh )is an intertwining operator, i.e., we have U g ρ ( h ) = ρ ( h ) U g for all h ∈ G n . By Schur’slemma, this forces U g to be a constant multiple of the identity, that is, we have U g = λI V for some λ = λ ( g ) ∈ C . Taking traces on both sides of (111) and using the fact that χ ρ ( h − gh ) = χ ρ ( g )for all h ∈ G n shows that λ = χ ρ ( g ), proving (110).Observe that for every unit vector e ∈ V we have X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ e (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V . Now we fix a unit vector e ∈ V . Using the relation (110), we write X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ e = X a ∈ W jN dim( ρ ) | G n | X h ∈ G n ρ ( h − gγ − a h ) γ ′ a ( x ) δ e = dim( ρ ) | G n | X a ∈ W jN X h ∈ G n ρ ( h − g ) ρ ( γ − a ) γ ′ a ( x ) δ ρ ( h ) e . Setting f h = ρ ( h ) e for all h ∈ G n and regarding f h as a (constant) function in H ( D, V ), we can rewrite the above as X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ e = dim( ρ ) | G n | X h ∈ G n ρ ( h − g ) X a ∈ W jN ρ ( γ a ) − γ ′ a ( x ) δ f h ( x )= dim( ρ ) | G n | X h ∈ G n ρ ( h − g )( L Nδ,ρ f h )( x ) . Hence, since ρ is unitary with respect to k · k V , we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ e (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) V ≤ dim( ρ ) | G n | X h ∈ G n kL Nδ,ρ f h ( x ) k V ≤ dim( ρ ) max h ∈ G n kL Nδ,ρ f h ( x ) k V . Invoking Proposition 6.1 yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X a ∈ W jN χ ρ ( gγ − a ) γ ′ a ( x ) δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ dim( ρ ) max h ∈ G n kL Nδ,ρ f h k ∞ , Λ ≪ dim( ρ ) e − cN , which concludes the proof. (cid:3) Proof of Theorem 1.4.
We can now prove Theorem 1.4. Let Γ be a non-elementary Schottky group, let Γ n be a family of finite-index normal subgroups, andassume that the G n ’s form a family of two-sided ε -expanders. Fix some absoluteconstants A > c >
0. We want to show that there exists η > ε , c , A and the Schottky data of Γ such that for all n and all irreducible representations( ρ, V ) of G n whose dimension is bounded below by c | G n | A , the twisted Selberg zetafunction Z Γ ( s, ρ ) has no zeros in the half-plane Re( s ) ≥ δ − η . For the remainder of this proof, we fix n and a non-trivial, irredicible representation( ρ, V ) of G n . By the result of Magee–Naud [29, Theorem 1.8], we know that thereexist constants η Γ > T Γ > every finite-index subgroup Γ ′ < Γ the Selberg zeta function Z Γ ′ ( s ) has no zeros in(112) { Re( s ) ≥ δ − η Γ , | Im( s ) | ≥ T Γ } . Thanks to the Venkov–Zograf formula (16), we conclude that Z Γ ( s, ρ ) does not vanishin the region (112) for any of the irreducible non-trivial representations of G n . Thusit suffices to show that there exists η > ε and the Schottky dataof Γ such that Z Γ ( s, ρ ) has no zero in R η ,T Γ := { Re( s ) ≥ δ − η , | Im( s ) | < T Γ } . Theorem 1.4 would then follow with η = min { η , η Γ } . By the arguments used in § s ∈ R η ,T Γ there exists a positive integer N such thatthe norm of L Ns,ρ : H ( D, V ) → H ( D, V ) satisfies kL Ns,ρ k < . We will obtain an estimate for kL Ns,ρ k through an estimate on the Hilbert–Schmidtnorm of L Ns,ρ . Recall that the Hilbert–Schmidt norm of a compact operator A on aseparable Hilbert space H is defined by k A k = ∞ X k =1 µ k ( A ) , where µ k ( A ) are the singular values of A . Recall that the singular values of A are theeigenvalues of the positive self-adjoint compact operator √ A ∗ A . Moreover, the largestsingular value µ ( A ) corresponds to the operator norm of A , µ ( A ) = k A k = sup v ∈ H, k v k H =1 k Av k H . In particular, we have k A k ≤ k A k HS . Furthermore, recall that the regular representation of G n is the representation R n : G n → U( L ( G n ))defined by R n ( g ) ϕ ( x ) = ϕ ( g − x ) . for all g ∈ G n , x ∈ G n and ϕ ∈ L ( G n ) . We can now state the main result needed tocomplete the proof of Theorem 1.4.
Lemma 7.3 (Hilbert–Schmidt norm) . For every finite-dimensional, unitary represen-tation ( ρ, V ) of Γ and all s ∈ C , we have the identity (113) kL Ns,ρ k = m X j =1 X a , b ∈ W jN χ ρ ( γ b γ − a ) Z D j γ ′ a ( z ) s γ ′ b ( z ) s B D ( γ a ( z ) , γ b ( z ))dvol( z ) , where χ ρ ( · ) = tr V ( ρ ( · )) is the character associated to ρ . In particular, for the regularrepresentation R n we have (114) kL Ns,R n k = | G n | m X j =1 X a , b ∈ W jN p n ( γ a )= p n ( γ b ) Z D j γ ′ a ( z ) s γ ′ b ( z ) s B D ( γ a ( z ) , γ b ( z ))dvol( z ) . Here, B D ( z, w ) denotes the Bergman kernel of the classical Bergman space H ( D ) . Inparticular, we have the bound (115) B D ( z, w ) ≤ π dist( z, ∂D )dist( w, ∂D ) for all z, w ∈ D , where dist denotes the euclidean distance. Before we prove Lemma 7.3 let us see how it can be used to finish the proof of Theorem1.4. In what follows we write s = σ + it , where σ and t are the real and imaginaryparts of s ∈ C respectively, and we assume that σ ∈ [0 , δ ] . Moreover, the c κ ’s appearingbelow are positive constants which are independent of n and s , that is, they are allowedto depend only on ε , c , A and the Schottky data of Γ.Using the bounded distortion property (23) together with the fact that the derivatives γ ′ a are positive real numbers on the real line, we have | γ ′ a ( z ) s | ≤ c e c | t | k γ ′ a k σ ∞ ,D j for all j ∈ { , . . . , m } , all z ∈ D j and all a ∈ W jN . Using the uniform hyperbolicityproperty (21), we obtain furthermore(116) | γ ′ a ( z ) s | ≤ c e c | t | e c N ( δ − σ ) k γ ′ a k δ ∞ ,D j . Notice also that for all j ∈ { , . . . , m } , N ≥ z ∈ D j and all a , b ∈ W jN , the images γ a ( z ) and γ b ( z ) are uniformly bounded away from the boundary ∂D . Together withthe bound in (115), this gives(117) B ( γ a ( z ) , γ b ( z )) ≤ c . We can now use Lemma 7.3 together with (116) and (117) to obtain the estimate(118) kL Ns,R n k ≤ c e c | t | e c N ( δ − σ ) | G n | m X j =1 X a , b ∈ W jN p n ( γ a )= p n ( γ b ) k γ ′ a k δ ∞ ,D j k γ ′ b k δ ∞ ,D j Notice that the inner sum on the right can be rewritten as X a , b ∈ W jN p n ( γ a )= p n ( γ b ) k γ ′ a k δ ∞ ,D j k γ ′ b k δ ∞ ,D j = X g ∈ G n X a ∈ W jN p n ( γ a )= g k γ ′ a k δ ∞ ,D j . By Corollary 7.2, there are constants c > c > n , for all N ≥ c log | G n | , and for all g ∈ G n , X a ∈ W jN p n ( γ a )= g k γ ′ a k δ ∞ ,D j ≤ c | G n | . Thus, choosing N n = ⌈ c log | G n |⌉ , we obtain X a , b ∈ W jNn p n ( γ a )= p n ( γ b ) k γ ′ a k δ ∞ ,D j k γ ′ b k δ ∞ ,D j ≤ c | G n | . Inserting this bound into (118) gives(119) kL N n s,R n k ≤ c e c | t | e c N n ( δ − σ ) ≤ c e c | t | | G n | c ( δ − σ )3 Recall that the regular representation R n decomposes as a direct sum over the irre-ducible representations of G n , R n = M ξ ∈ b G n dim( ξ ) ξ. Combining this with the identity for the Hilbert-Schmidt norm in (113), we deducethat kL N n s,R n k = X ξ ∈ b G n dim( ξ ) kL N n s,ξ k . In particular, since ρ is assumed to be irreducible, we obtain(120) kL N n s,ρ k ≤ kL N n s,ρ k ≤ kL N n s,R n k dim( ρ ) . Now we exploit the fact that the dimension of ρ is large: inserting the bounddim( ρ ) ≥ c | G n | A into (120) and using (119) yields kL N n s,ρ k ≤ c − c e c | t | | G n | c (cid:16) δ − σ − Ac (cid:17) = c e c | t | | G n | c (cid:16) δ − σ − Ac (cid:17) . We can now set η = A c . Notice that η depends only on ε , A and the Schottky dataof Γ. With this choice, we get for all s ∈ R η ,T Γ the bound kL N n s,ρ k ≤ c e c T Γ | G n | c (cid:16) η − Ac (cid:17) = c | G n | − A . Assuming that the size of the group G n is sufficiently large (that is, larger than c − /A ),this gives kL N n s,ρ k < s ∈ R η ,T Γ , as desired. Notice that we may assume without loss of generalitythat the size of G n is large enough. Indeed, for every K > X = Γ \ H of degree less than K , up to isometry. By Naud’s result[26], each of these coverings has a positive “spectral gap” η . Letting η ′ > η obtained above by min { η , η ′ } and the statement of Theorem 1.4 holdstrue without any additional assumption on the size of the covering groups G n . Proof of Lemma 7.3.
The arguments we use here are similar to those contained in [29,Lemma 4.7] and [35, Proposition 5.5]. Let D , . . . , D m be the Schottky disks arisingfrom the construction of Γ. For every j ∈ { , . . . , m } let c j ∈ R and r j > D j respectively. One verifies that the family of functions ϕ j,ℓ ( z ) = s ℓ + 1 πr j (cid:18) z − c j r j (cid:19) ℓ , z ∈ D j , ℓ ∈ N forms an orthonormal basis for the classical Bergman space H ( D j ). Using this basis,we can compute the Bergman (reproducing) kernel of this space as(121) B D j ( z, w ) = ∞ X ℓ =0 ϕ j,ℓ ( z ) ϕ j,ℓ ( w ) = r j π (cid:16) r j − ( z − c j )( w − c j ) (cid:17) , where the series on the left converges absolutely for any two points z and w in theinterior of D j . The Bergman kernel of the space H ( D ) is then given by B D ( z, w ) = ( B D j ( z, w ) if both z and w belong to D j z and w belong to different disks . From the explicit formula in (121) it follows that B D ( z, w ) ≤ π dist( z, ∂D )dist( w, ∂D )for every z, w ∈ D . Now let ( ρ, V ) be a unitary representation of Γ and let e , . . . , e d (where d = dim( ρ )) be an orthonormal basis for the representation space V . Then thefamily of functions Ψ ℓ,k with 1 ≤ k ≤ d and ℓ ∈ N given byΨ ℓ,k ( z ) = ( ϕ j,ℓ ( z ) e k if z ∈ D j , V -valued Bergman space H ( D, V ). We canthen compute the Hilbert–Schmidt norm as kL Ns,ρ k = d X k =1 X ℓ ∈ N kL Ns,ρ Ψ ℓ,k k = d X k =1 X ℓ ∈ N m X j =1 X a , b ∈ W jN h ρ ( γ − a ) e k , ρ ( γ − b ) e k i V × Z D j γ ′ a ( z ) s γ ′ b ( z ) s ϕ j,ℓ ( γ a ( z )) ϕ j,ℓ ( γ b ( z )) dvol( z ) . We can now interchange sums. Since ρ is a unitary representation, we have d X k =1 h ρ ( γ − a ) e k , ρ ( γ − b ) e k i V = tr V ( ρ ( γ b γ − a )) = χ ρ ( γ b γ − a ) . Using this together with (121), we obtain the expression kL Ns,ρ k = m X j =1 X a , b ∈ W jN χ ρ ( γ b γ − a ) Z D j γ ′ a ( z ) s γ ′ b ( z ) s B D ( γ a ( z ) , γ b ( z )) dvol( z ) , which is precisely what was claimed in (113). We can now specialize this to the regularrepresentation R n . Note that the character of the regular representation χ R n ( γ ) is notequal to zero if and only if p n ( γ ) is the identity in G n , in which case it is equal to | G n | .In particular, χ R n ( γ b γ − a ) = ( | G n | if p n ( γ a ) = p n ( γ b ) , kL Ns,R n k = | G n | m X j =1 X a , b ∈ W jN p n ( γ a )= p n ( γ b ) Z D j γ ′ a ( z ) s γ ′ b ( z ) s B ( γ a ( z ) , γ b ( z )) dvol( z ) . This completes the proof of Lemma 7.3. (cid:3) References [1] A. Beardon,
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