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Let X be a finite-area hyperbolic surface, that is, a smooth surface with Riemannian metric of constant curvature 1. In this paper, all hyperbolic − surfaces are assumed complete and orientable. The Laplacian operator ∆X on L2(X) has spectrum in [0, ). The bottom of the spectrum is always a ∞ discrete eigenvalue at 0 that is simple if and only if X is connected. If X is non-compact, which is the first focus of our paper, then the 1 spectrum of ∆X in [0, 4 ) consists of finitely many discrete eigenvalues and 1 the spectrum is absolutely continuous in [ 4 , ) [LP81]. As such, the spectral ∞ 1 gap between 0 and the rest of the spectrum has size at most 4 . We are interested in the size of this spectral gap for random surfaces. The random model we use is the random covering model from [MN20, MNP20, MN21]. For any n N, the space of degree n Riemannian covering spaces ∈ of X is a finite set that we equip with the uniform probability measure. We say that a family of events, depending on n, holds asymptotically almost surely (a.a.s.) if they hold with probability tending to one as n . We → ∞ also note that any eigenvalue of the Laplacian on X will be an eigenvalue of any cover of X, with at least as large multiplicity. Therefore these ‘old eigenvalues’ must always be taken into account. The first theorem of the paper is the following.

Theorem 1.1. Let X be a finite-area non-compact hyperbolic surface. Let Xn denote a uniformly random degree n covering space of X. For any ǫ> 0, a.a.s. 1 1 spec(∆ ) 0, ǫ = spec(∆ ) 0, ǫ Xn ∩ 4 − X ∩ 4 −     and the multiplicities on both sides are the same.

This theorem is the analog, for finite-area non-compact hyperbolic sur- faces of Friedman’s theorem [Fri08] (formerly Alon’s conjecture [Alo86]) stat- ing that random d-regular graphs have almost optimal spectral gaps. Fried- man also proposed in [Fri03] that a variant of Alon’s conjecture should hold for random covers of any fixed finite graph, and this extended conjecture was recently proved by Bordenave and Collins [BC19]. If X has Euler characteristic χ(X)= 1, then a result of Otal and Rosas − [OR09, Thm. 2] states that spec(∆ ) 0, 1 is just the multiplicity one X ∩ 4 eigenvalue at 0. Taking such an X as the base surface in Theorem 1.1 we 
therefore obtain the following corollary.

2 Corollary 1.2. There exist connected finite-area non-compact hyperbolic surfaces X with χ(X ) and i i → −∞ 1 inf (spec(∆ ) (0, )) . Xi ∩ ∞ → 4 In fact one can take χ(X )= i. i − The corresponding question for closed surfaces X is as follows. If X is closed then the spectrum of ∆X consists of eigenvalues

0= λ (X) λ (X) λ (X) 0 ≤ 1 ≤···≤ i ≤··· with λ (X) as i . Suppose that X are a sequence of closed i → ∞ → ∞ i hyperbolic surfaces with genera g(X ) . Huber proved in [Hub74] that i →∞ in this case, 1 lim sup λ (X ) . 1 i ≤ 4 1 Therefore 4 is an asymptotically optimal lower bound for λ1(Xi). In the same scenario, it was conjectured in [Bus78] that lim λ (X ) 0. This i→∞ 1 i → was corrected in [Bus84] where it was put forward that ‘probably’ there exist a sequence of Xi as above with 1 lim λ1(Xi)= . i→∞ 4 See [WX18, Conj. 5], [Wri20, Problem 10.3] for more recent iterations of 1 the question of whether λ1(Xi) can go to 4 as the genus tends to infinity. We are able to resolve this question here by combining Corollary 1.2 with the ‘handle lemma’ of Buser, Burger, and Dodziuk [BBD88]; see Brooks and Makover [BM01, Lemma 1.1] for the extraction of the lemma. We obtain

Corollary 1.3. There exist closed hyperbolic surfaces Xi with genera g(X ) and λ (X ) 1 . In fact one can take g(X )= i. i →∞ 1 i → 4 i 1.1 Prior work The analog of Theorem 1.1 was proved for conformally compact infinite area hyperbolic surfaces by the second named author and Naud in [MN21], following a previous work with an intermediate result [MN20]. The reader should see however [MN21] for details because the spectral theory is more subtle for infinite area surfaces and moreover the results in [MN21] go beyond statements about L2 eigenvalues.

3 The very first uniform spectral gap for certain combinatorial models of random surfaces (the ‘Brooks-Makover’ models), yielding both finite-area non-compact and closed surfaces, appears in work of Brooks and Makover [BM04]. This spectral gap is non-explicit. The analog of Theorem 1.1 for closed hyperbolic surfaces is still unknown. The best known result in the random cover model is due to the second named author, Naud, and Puder [MNP20], building on [MP20], that gives an a.a.s. relative1 spectral gap of size 3 ǫ for random covers of a closed hyperbolic 16 − surface. The fact that a Weil-Petersson random closed hyperbolic surface enjoys spectral gap of size 3 ǫ with probability tending to one as genus 16 − was proved independently by Wu and Xue [WX21] and Lipnowski → ∞ and Wright [LW21]. The first uniform spectral gap of size 0.0024 for ≈ Weil-Petersson random closed surfaces was proved by Mirzakhani in [Mir13]. Prior to the current work, the first named author proved in [Hid] that Weil- α 1 Petersson random surfaces of genus g with O(g ) cusps, with α < 2 , have an explicit positive uniform spectral gap depending on α as g , which → ∞ coincides (for any fixed ǫ> 0) with 3 ǫ if α = 0. 16 − The fact that all models of random closed hyperbolic surfaces are cur- rently stuck at λ = 3 ǫ is an obstacle to proving Corollary 1.3 by the 1 16 − more direct method of proving that random closed surfaces of large genus 3 have almost optimal spectral gaps. This 16 barrier bears explanation; it 3 also appears in the famous 16 Theorem of Selberg [Sel65]. Very roughly speaking, this barrier corresponds to fine control of probability events con- cerning simple closed geodesics in random surfaces and not having as fine control on non-simple geodesics. This splitting of geodesics into simple and non-simple goes back to work of Broder and Shamir [BS87] in the graph 3 theoretic setting. The appearance of 16 in [Sel65] is for different reasons relating to Selberg’s use of Weil’s bounds [Wei48] on Kloosterman sums. The previous best results towards Corollary 1.2, just for the case of χ(Xi) , all came from arithmetic hyperbolic surfaces. The records → −∞ 3 were held by Selberg [Sel65] (spectral gaps of size 16 = 0.1875), Gelbart and 3 Jacquet [GJ78] (existence of spectral gaps larger than 16 ), Luo, Rudnick, and Sarnak [LRS95] (spectral gaps of size 171 0.218), Kim and Shahidi 784 ≈ [KS02b, KS02a] (spectral gaps of size 77 0.23765), and currently, Kim 324 ≈ and Sarnak [Kim03, Appendix 2] (spectral gaps of size 975 0.23804). 4096 ≈ The history of Corollary 1.3 began with McKean’s paper [McK72, McK74] where it was wrongly asserted that the first eigenvalue of any compact hyper- 1 bolic surface is at least 4 ; this was disproved by Randol in [Ran74] who actu- 1Here relative refers to disregarding eigenvalues of the base surface.

4 1 ally showed that there can be arbitrarily many eigenvalues in (0, 4 ). Buser 3 proved in [Bus84] using Selberg’s 16 theorem and the Jacquet-Langlands machinery [JL70] that there exist closed hyperbolic surfaces Xi with gen- era g(X ) and λ (X ) 3 . Buser, Burger, and Dodziuk proved in i → ∞ 1 i ≥ 16 [BBD88] that one can get the slightly weaker result λ (X ) 3 but without 1 i → 16 using Jacquet-Langlands. This turns out to be instrumental in the current work; see Lemma 8.1 below. Following these works, all progress towards Corollary 1.3 ran parallel to the previously discussed progress to Corollary 1.2. In particular, the previous best known result towards Corollary 1.3 comes from combining the Kim-Sarnak bound with either Jacquet-Langlands or [BBD88] to obtain the 1 975 first part of Corollary 1.3 with 4 replaced by 4096 .

1.2 Overview of proof The proof of Theorem 1.1 follows the following strategy. We view

X =Γ H \ where Γ is a discrete torsion free subgroup of PSL2(R) and H is the hyper- bolic upper half plane. We explain in 2 that the random degree n covers § of X are parameterized by

φ Hom(Γ,S ) X ∈ n 7→ φ where φ is a uniformly random homomorphism of the free group Γ into the symmetric group Sn. Because we disregard eigenvalues and eigenfunctions 2 lifted from X, we restrict our attention to the space Lnew(Xφ) of functions that are orthogonal to all lifts of L2 functions from X. The strategy is then, 1 for any s0 > 2 , to asymptotically almost surely produce a bounded resolvent operator R (s) : L2 (X ) H2 (X ) Xφ new φ → new φ where H2 (X ) L2 (X ) is a suitable Sobolev space (see 2.2) such that new φ ⊂ new φ § for any s [s , 1] the identity ∈ 0 ∆ s(1 s) R (s)=Id 2 , Xφ − − Xφ Lnew(Xφ) makes sense and holds true; this will
forbid new eigenvalues in [0,s (1 s )]. 0 − 0 The way we build our resolvent is by a parametrix construction. The fundamental structure of parametrix construction is to produce a bounded operator M (s) : L2 (X ) H2 (X ) φ new φ → new φ

5 such that ∆ s(1 s) M (s)=Id 2 + L (s) Xφ − − φ Lnew(Xφ) φ where we aim to prove Lφ(s) has
norm less than one as a bounded operator 2 on Lnew(Xφ) so that we can obtain our bounded resolvent

−1 M 2 L RXφ (s)= φ(s) IdLnew(Xφ) + φ(s) .   Often in a parametrix construction one also wants the L term to be compact: this will also be the case here and turns out to be essential for the application of random operator results. The way we build Mφ(s) is by patching together a ‘cuspidal parametrix’ Mcusp φ (s) based on a model resolvent in the cusp and an an interior parametrix Mint φ (s) that only depends on the localization of its argument to a compact part of Xφ. We then let M Mint Mcusp φ(s)= φ (s)+ φ (s) and we get a resulting splitting

L Lint Lcusp φ(s)= φ (s)+ φ (s).

In 4 we show that for any φ, the term Mcusp(s) can be designed so that § φ Lcusp(s) 1 (or any small number), so that it will not essentially interfere k φ k≤ 5 with our plans of obtaining L (s) < 1. The ability to do this is a particular k φ k feature of the geometry of hyperbolic cusps. Mint The term φ (s) is based on averaging the resolvent kernel of the hy- perbolic plane over the fundamental group of Γ (suitably twisting by φ) to 2 obtain an integral operator on Lnew(Xφ). The problem with this is that the averaging will not obviously converge, so we have to multiply the hyperbolic resolvent kernel by a radial cutoff that localizes to radii T + 1 to get a ≤ priori convergence for all s ( 1 , 1]. This gives us that Mint(s) is bounded ∈ 2 φ (Lemma 5.5). Lint The effect of this cutoff is that the error term φ (s) is an integral operator with smooth kernel. We prove that we can unitarily conjugate Lint φ (s) to a (s) φ(γ) γ ⊗ γX∈Γ acting on L2(F ) V 0, where F is a Dirichlet fundamental domain for Γ ⊗ n and V 0 is the standard n 1 dimensional irreducible representation of S . n − n

6 2 The aγ (s) are compact operators on L (F ) and there are only finitely many γ Γ for which aγ(s) is non-zero. ∈ r If instead, the aγ(s) were elements of End(C ) for some fixed finite r, because Γ was free we would now be exactly in the situation to apply the breakthough results of Bordenave and Collins from [BC19]. These random operator results, combined with a linearization trick of Pisier [Pis18], would tell us that for any ǫ> 0, a.a.s.

Lint(s) a (s) ρ (γ) + ǫ (1.1) k φ k≤ γ ⊗ ∞ γ∈Γ X

2 where ρ∞ :Γ End(ℓ (Γ)) is the right regular representation. Because the → aγ (s) are in reality compact operators on Hilbert spaces we can approximate by finite rank operators to the same effect. The key point (which is a sticking point in other approaches to this problem using e.g. transfer operators) is that we understand the operator in the right hand side of (1.1) well: it can be unitarily conjugated to an operator on L2(H) that is the composition of multiplication with a cutoff (with norm 1) and an integral operator with real-valued radial kernel that ≤ localizes to radii in [T, T + 1]. The key is that as this latter operator is self-adjoint we can use the theory of the Selberg transform to estimate its norm in Lemma 5.2. By choosing T sufficiently large in the beginning, we can force the norm in the 1 right hand side of (1.1) to be as small as we like, given any s0 > 2 , for all s s . ≥ 0 Assembling these arguments, for any s s > 1 , a.a.s. 1 + L (s) can be ≥ 0 2 φ inverted which rules out X having a new eigenvalue at s(1 s). To be able φ − to get this result for all s s with probability tending to one, we make ≥ 0 sure that L (s) 3 at a fine enough net of s [s , 1] and then use a k φ k ≤ 5 ∈ 0 deviations estimate (Lemma 6.1) to show (deterministically) that Lφ(s) 1 k k fluctuates at most by 5 from point to point.

Acknowledgments We thank A. Gamburd, C. Bordenave, B. Collins, F. Naud, D. Puder, and P. Sarnak for conversations about this work. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 949143).

7 2 Random covers

2.1 Construction Let X be a finite area non-compact connected hyperbolic surface. We view X as X =Γ H \ where H = x + iy : x,y R,y> 0 { ∈ } with metric dx2 + dy2 y2 where Γ is a discrete torsion-free subgroup of PSL2(R) that acts, via M¨obius transformations, by orientation preserving isometries on H. def For n N let [n] = 1,...,n and S denote the group of permutations ∈ { } n of [n]. For any homomorphism φ Hom(Γ,S ) we construct a hyperbolic ∈ n surface as follows. Let Γ act on H [n] by × def γ(z,x) = (γz,φ(γ)[x]) and let def X = Γ (H [n]) . φ \φ × If we choose φ uniformly at random in Hom(Γ,Sn), the resulting Xφ is a uniformly random degree n covering space of X. Note that Γ is a free group freely generated by some γ , . . . , γ Γ 1 d ∈ and choosing φ is the same as choosing

def σi = φ(γi), i = 1,...,d independently and uniformly at random in Sn. def 2 0 Let Vn = ℓ ([n]) and Vn Vn the subspace of functions on [n] with zero ⊂ 2 mean. The representation of Sn on ℓ ([n]) is its standard representation by 0 0-1 matrices and the subspace Vn is an irreducible subspace of dimension (n 1): we write − ρ0 :Γ End(V 0) n → n for the random representation of Γ induced by the random φ.

8 2.2 Function spaces We define 2 Lnew(Xφ) to be the space of functions that are orthogonal to all pullbacks of L2 func- tions from X. We have

L2(X ) = L2 (X ) L2(X). φ ∼ new φ ⊕ This induces a multiplicity respecting inclusion spec(∆ ) spec(∆ ). All X ⊂ Xφ other spectra of ∆Xφ (with multiplicities) arise from eigenfunctions in the 2 subspace Lnew(Xφ). Let F denote a Dirichlet fundamental domain for X, that is, for some fixed o H, ∈ def F = z H : d(o, z) < d(z, γo) . { ∈ } γ∈Γ\\{id} ∞ H 0 0 H Let C ( ; Vn ) denote the smooth Vn -valued functions on . There is an isometric linear isomorphism between

C∞(X ) L2 (X ) φ ∩ new φ 0 H and the space of smooth Vn -valued functions on satisfying

0 f(γz)= ρn(γ)f(z) (2.1) for all γ Γ, with finite norm ∈ 2 def 2 f 2 = f(z) 0 dH < k kL (F ) k kVn ∞ ZF ∞ H 0 ∞ H 0 We will denote Cφ ( ; Vn ) C ( ; Vn ) for these functions. Under this ⊂ ∞ 2 isomorphism, the Laplacian on C (Xφ) Lnew(Xφ) is intertwined with the ∞ H 0 ∩ Laplacian that acts on Cφ ( ; Vn ) in the obvious way (this can be defined 0 by choosing any basis of Vn and letting the Laplacian act coordinatewise 0 ∞ H 0 on Vn -valued functions). The completion of Cφ ( ; Vn ) with respect to 2 2 H 0 L (F ) is denoted by Lφ( ; Vn ); the isomorphism above extends to one k • k 2 2 H 0 between Lnew(Xφ) and Lφ( ; Vn ). ∞ H 0 ∞ H 0 Let Cc,φ( ; Vn ) denote the elements of Cφ ( ; Vn ) that are compactly supported when restricted to F¯ (in other words, compactly supported mod- ulo Γ).

9 We also consider the following Sobolev spaces. Let H2(H) denote the ∞ H completion of Cc ( ) with respect to the norm

2 def 2 2 f 2 H = f 2 H + ∆f 2 H . k kH ( ) k kL ( ) k kL ( ) 2 H 0 ∞ H 0 We let Hφ( ; Vn ) denote the completion of Cc,φ( ; Vn ) with respect to the norm 2 def 2 2 f 2 H 0 = f L2(F ) + ∆f L2(F ). k kHφ( ;Vn ) k k k k 2 ∞ We let H (Xφ) denote the completion of Cc (Xφ) with respect to the norm

2 def 2 2 f 2 = f 2 + ∆f 2 . k kH (Xφ) k kL (Xφ) k kL (Xφ) 2 2 Viewing H (Xφ) as a subspace of L (Xφ) in the obvious way, we let

H2 (X ) def= H2(X ) L2 (X ). new φ φ ∩ new φ 2 Similarly to before there is an isometric isomorphism between Hnew(Xφ) 2 H 0 and Hφ( ; Vn ) that intertwines the two relevant Laplacian operators.

3 Background random matrix theory

In this section we give an account of the breakthrough result of Bordenave and Collins [BC19] and its amplification through a linearization trick that appears in Pisier [Pis18, Cor. 14] (see also, historically, [Pis96, HT05], and [BC19, §6]). Let std : S End(V 0) denote the linear action of S on V 0. We let n n → n n n ρ : Γ End(ℓ2(Γ)) denote the right regular representation of Γ. We are ∞ → going to use the following direct consequence of [BC19, Thm. 2].

Theorem 3.1 (Bordenave-Collins). Suppose that r N and a0, a1, . . . ad ∗ ∈ ∈ Matr×r(C). Suppose a0 = a0. Then for any ǫ> 0, with probability tending to one as n , we have →∞ d ∗ −1 a Id 0 + a std (σ )+ a std (σ ) 0 ⊗ Vn i ⊗ n i i ⊗ n i i=1 Cr 0 ⊗Vn X d 2 ∗ −1 a0 Idℓ (Γ) + ai ρ∞(γi)+ ai ρ∞(γi ) + ǫ. ≤ ⊗ ⊗ ⊗ 2 i=1 Cr⊗ℓ (Γ) X r 0 The norm on the top is the operator norm on C V . The norm on the ⊗ n bottom is the operator norm on Cr ℓ2(Γ). ⊗

10 0 Remark 3.2. Note in the above that stdn(σi)= ρn(γi). This will be combined with the following result of Pisier [Pis18, Cor. 14 and following remark].

Proposition 3.3 (Pisier). The result of Theorem 3.1 implies that for any R, k N, any non-commutative polynomials ∈

P1, P2,...,Pk in the variables σ ,...,σ ,σ−1, ,σ−1 with complex coefficients, and any 1 d 1 · · · d a , . . . , a Mat (C) we have the following. For any ǫ> 0, with proba- 1 k ∈ R×R bility tending to one as n , we have →∞ k a std (P (σ ,...,σ ,σ−1,...,σ−1)) (3.1) i ⊗ n i 1 d 1 d i=1 CR 0 ⊗Vn X k −1 −1 ai ρ∞(Pi(γ1, . . . , γd, γ1 , . . . , γd )) + ǫ. ≤ ⊗ 2 i=1 CR⊗ℓ (Γ) X

This type of ‘linearization’ is obviously extremely powerful and has played a role in many of the major breakthroughs in random operators in recent years [HT05, CM14, BC19]. Notice that one way to obtain an operator as in (3.1) is as a ρ0 (γ) γ ⊗ n Xγ∈Γ where all a Mat (C) and there are only finitely many non-zero a . γ ∈ R×R γ Therefore we obtain combining Theorem 3.1 and Proposition 3.3 the follow- ing:

Corollary 3.4. For any r N and any finitely supported map γ Γ ∈ ∈ 7→ a Mat (C), for any ǫ > 0, with probability tending to one as n γ ∈ r×r → ∞ we have

a ρ0 (γ) a ρ (γ) + ǫ. γ ⊗ n ≤ γ ⊗ ∞ γ∈Γ Cr 0 γ∈Γ Cr 2 X ⊗Vn X ⊗ℓ (Γ)

No originality is claimed here: Corollary 3.4 is essentially already con- tained in [BC19] and in fact, [BC19, Thm. 3] is the version of this corollary when all ai are scalar.

11 4 Cusp parametrix

To simplify notation, we assume there is just one cusp of the finite-area non compact hyperbolic surface X. This does not affect the arguments. This single cusp can be identified with

def = (1, ) S1 C ∞ × with the metric dr2 + dx2 ; (4.1) r2 here the x coordinate is in S1 def= R/Z and the r coordinate is in [1, ). In ∞ the following, χ+,χ− : [0, 1] will be functions that are identically zero C C C → in a neighborhood of 1 S1, identically equal to 1 in a neighborhood of { }× S1, and such that {∞} × + − − χC χC = χC . (4.2) We will extend both these functions by zero to functions on X, and define

def χ± = χ± π : X [0, 1] C,φ C ◦ φ φ → where π : X X is the covering map. Indeed, the cusp of X splits in X φ φ → φ into several regions of the form

(1, ) (R/mZ) , (4.3) ∞ × with m N, and with the same metric (4.1). In these coordinates the ∈ covering isometry sends

π : (r, x + mZ) (r, x + Z). φ 7→ In particular, it preserves the r coordinate.

± Lemma 4.1. For any ǫ> 0, it is possible to choose χC as above so that for any φ χ+ , ∆χ+ ǫ. k∇ C,φk∞ k C,φk∞ ≤ + + Proof. Given ǫ> 0, let χC : [0, ) [0, 1] be a function such that χC (τ) + ∞ → ≡ 0 for τ in [0, 1], χC (τ) 1 for τ τ0, for some τ0 > 0, and such that we ≡ ≥ + have the following bounds for the derivatives of χC

+ ′ + ′′ ǫ sup (χC ) , sup (χC ) [0,∞) | | [0,∞) | |≤ 2

12 (this can be achieved by scaling some fixed cutoff). Let ′ be any cusp region of X as in (4.3). Using the change of coordi- C φ nates r = eτ we view ′ as C (0, ) R/mZ ∞ τ × with the metric (dτ)2 + e−2τ (dx)2; x being the coordinate in R/mZ. In these coordinates,

χ+ (τ,x + mZ)= [χ+]′(τ) ǫ k∇ C,φk | C |≤ and

∆χ+ (τ,x + mZ)= [χ+]′′(τ) [χ+]′(τ) ǫ | C,φ| | C − C |≤ − − as required. Finally, one can easily choose χC to be a function with χC (τ) 0 for τ τ and χ−(τ) 1 for τ 2τ . This will fulfill (4.2). ≡ ≤ 0 C ≡ ≥ 0 Let denote the subset of X that covers . It is convenient, to avoid Cφ φ C complicated discussions about Sobolev spaces, to extend to the parabolic C cylinder ˜ def= (0, ) S1 C ∞ × with the same metric (4.1), and let ˜ be the corresponding extension of Cφ Cφ (the union of extensions of (4.3) to (0, ) R/mZ). ∞ × Let H2( ˜ ) denote the completion of C∞( ˜ ) with respect to the given Cφ c Cφ norm 2 def 2 2 f 2 = f 2 + ∆f 2 . k kH k kL k kL ∞ The Laplacian ∆ = ∆ ˜ extends uniquely from C ( ˜φ) to a self-adjoint Cφ c C unbounded operator on L2( ˜ ) with domain H2( ˜ ). Cφ Cφ Lemma 4.2. For any f H2( ˜ ), we have ∆f,f 1 f 2. ∈ Cφ h i≥ 4 k k Proof. This is similar to [Mag15, Lemma 3.2]. It suffices to prove this for ˜ replaced by (0, ) (R/mZ) with the metric (4.1) i.e. with only one Cφ ∞ × connected component. Then changing coordinates to τ we are working in the region ( , ) (R/mZ) with the metric (dτ)2 + e−2τ (dx)2. The −∞ ∞ × corresponding volume form is e−τ dτ dx and the Laplacian is given by 2 τ ∂ −τ ∂ 2τ ∂ ∧ ∞ ∆ = e e e 2 . Now suppose f C (( , ) (R/mZ)). − ∂τ ∂τ − ∂θ ∈ c −∞ ∞ × We calculate ∂2 1 ∂2 e−τ/2∆eτ/2 = + e2τ −∂τ 2 4 − ∂θ2

13 so if g = e−τ/2f C∞(( , ) (R/mZ)) we have ∈ c −∞ ∞ × ∞ n ∆[f]fe¯ −τ dτ dx = [e−τ/2∆eτ/2] (g)gdτ ¯ dx ∧ ∧ Z Z−∞ Z0 1 ∞ n 1 ∞ n ggdτ¯ dx = ffe¯ −τ dτ dx. ≥ 4 ∧ 4 ∧ Z−∞ Z0 Z−∞ Z0 The inequality here used integrating by parts. The inequality obtained now extends to H2( ˜ ) by density of C∞(C˜ ) therein and continuity of Cφ c φ ∆f,f . h i Lemma 4.2 implies that the resolvent operator

def −1 2 2 R ˜ (s) = (∆ s(1 s)) : L ( ˜φ) H ( ˜φ) Cφ − − C → C 1 is a holomorphic family of bounded operators in Re(s) > 2 , each a bijection to their image. This gives an a priori bound for the resolvent: using

(∆ s(1 s)) R ˜ (s)f = f − − Cφ and Lemma 4.2 we obtain that for f L2( ˜ ) and s ( 1 , 1] ∈ Cφ ∈ 2 1 −1 R ˜ (s)f 2 s(1 s) f 2 (4.4) k Cφ kL ≤ 4 − − k kL   and as ∆R ˜ (s)f = f + s(1 s)R ˜ (s)f Cφ − Cφ we obtain for s ( 1 , ) ∈ 2 ∞

∆R ˜ (s)f 2 f 2 + s(1 s) R ˜ (s)f 2 k Cφ kL ≤ k kL − k Cφ kL s(1 s) 1+ − f 2 ≤ 1 s(1 s) k kL 4 − − ! 1 = f 2 . (4.5) 1 4s(1 s)k kL − − We now define the cusp parametrix as

Mcusp(s) def= χ+ R (s)χ− . (4.6) φ C,φ C˜φ C,φ

Here,

14 − 2 • (multiplication by) χC,φ is viewed as an operator from L (Xφ) to L2( ˜ ) by mapping first to L2( ) and then extending by zero. This Cφ Cφ is a bounded linear operator.

2 2 • R ˜ (s) is a bounded operator from L ( ˜φ) to H ( ˜φ). Cφ C C + 2 ˜ 2 • (multiplication by) χC,φ is viewed as a operator from H ( φ) to H (Xφ), + C using that χC,φ localizes to φ and then extending by zero. This opera- C + tor is bounded because derivatives of χC,φ are bounded and compactly supported.

Hence M cusp(s) : L2(X ) H2(X ) φ φ → φ is a bounded operator. The covering map extends in an obvious way to a covering map Cφ →C ˜φ ˜ that intertwines the two Laplacian operators. This, together with C → C − + the fact that multplication by χC,φ and χC,φ leave invariant the subspaces of functions lifted through the covering map, one sees that

M cusp(s)(L2 (X )) H2 (X ). φ new φ ⊂ new φ + − − Because χC,φχC,φ = χC,φ,

cusp − + − − cusp (∆ s(1 s))M (s)= χ + [∆,χ ]R ˜ (s)χ = χ + L (s) (4.7) − − φ C,φ C,φ Cφ C,φ C,φ φ where def Lcusp(s) = [∆,χ+ ]R (s)χ− . φ C,φ C˜φ C,φ − + 2 Here again we view χ and R ˜ (s) as above, and [∆,χ ] : H ( ˜φ) C,φ Cφ C,φ C → 2 Lcusp 2 L (Xφ). This means that φ (s) is an operator on L (Xφ). By similar + arguments to before, using that [∆,χC,φ] only involves radial derivatives + (since χC,φ is radial), we obtain

Lcusp(s) L2 (X ) L2 (X ). φ new φ ⊂ new φ 1
Lemma 4.3. Given s0 > 2 , there exists C(s0) > 0 such that for s [s0, 1], Lcusp 2 ∈ the operator φ (s) is a self-adjoint, bounded operator on L (Xφ) with operator norm

cusp + + L (s) 2 (∆χ ) + 2 χ C(s ). k φ kL ≤ k C,φ k∞ k∇ C,φk∞ 0   15 Proof. As an operator on H2( ˜ ) Cφ [∆,χ+ ]=(∆χ+ ) 2( χ+ ) . C,φ C,φ − ∇ C,φ ·∇ The first summand is a multiplication operator; for f H2( ˜ ) we have ∈ Cφ + + (∆χ )f 2 (∆χ ) f 2 (4.8) k C,φ kL ≤ k C,φ k∞k kL and by Schwarz inequality if f 2 1 then k kH ≤

+ 2 + 2 ( χC,φ) f L χC,φ ∞ f L k ∇ ·∇ k ≤ k∇ k k∇ k 1 1 1 = χ+ ∆f,f 2 χ+ ∆f 2 f 2 k∇ C,φk∞h i ≤ k∇ C,φk∞k k k k χ+ . (4.9) ≤ k∇ C,φk∞ + The two estimates (4.8), (4.9) hence show that [∆,χC,φ] has norm bounded + + 2 ˜ 2 by (∆χC,φ) ∞ + 2 χC,φ ∞ as a map H ( φ) L (Xφ). Since multipli- k −k k∇ k 2 2 C → cation by χ has norm 1 from L to L , and R ˜ (s) has norm C(s0) C,φ ≤ Cφ ≤ from L2( ˜ ) to H2( ˜ ) by (4.4) and (4.5), we obtain the stated result. Cφ Cφ We can now obtain the following key proposition by combining Lemmas 4.1 and 4.3. 1 + − Proposition 4.4. Given any s0 > 2 , we can choose χC and χC (depending on s0) such that for any n N and φ Hom(Γ,Sn) and any s [s0, 1], we ∈ Lcusp ∈ 2 ∈ have for the operator norm of φ (s) on L (Xφ)

cusp 1 L (s) 2 . k φ kL ≤ 5 − + We assume that χC and χC have now been fixed to satisfy Proposition 4.4 in the rest of the paper. Hence all constants may depend on these functions.

5 Interior parametrix

5.1 Background on spectral theory of hyperbolic plane Resolvent Let

2 2 RH(s) : L (H) L (H), → def −1 RH(s) = (∆H s(1 s)) . − −

16 1 H Since spec(∆ ) [ 4 , ) [Bor16, Thm. 4.3], this is well defined as a bounded ⊂ ∞1 operator for Re(s) > 2 . Given x,y H, we write r = dH(x,y) where dH is hyperbolic distance ∈ and r σ def= cosh2 . 2   The operator RH(s) is an integral operator with kernel [Bor16, Prop. 4.2]

1 1 ts−1(1 t)s−1 RH(s; x,y)= − dt. (5.1) 4π (σ t)s Z0 − For σ> 1 and t (0, 1) we have ∈ ∂ ts−1(1 t)s−1 r r ts−1(1 t)s−1 − = s sinh cosh − , ∂r (σ t)s − 2 2 (σ t)s+1 −     − ∂ ts−1(1 t)s−1 t(1 t) ts−1(1 t)s−1 − = log − − , ∂s (σ t)s (σ t) (σ t)s −  −  − ∂2 ts−1(1 t)s−1 r r ts−1(1 t)s−1 − = sinh cosh − ∂s∂r (σ t)s − 2 2 (σ t)s+1 − −   t(1  t) 1+ s log − . · (σ t)   −  Each of these are smooth in (s,r,t) [ 1 , 1] [1, ) (0, 1). Because for ∈ 2 × ∞ × s, r in a fixed compact set of [ 1 , 1] [1, ), these all have absolute values 2 × ∞ majorized by integrable functions of t (0, 1), we can interchange derivatives ∈ and integrals to get corresponding bounds for RH as follows. We define

def RH(s; r) = RH(s; x,y).

Firstly, there is a constant C > 0 such that for r 1 and s [ 1 , 1]we have 0 ≥ ∈ 2

∂RH −sr0 RH(s; r ) , (s; r ) Ce . (5.2) | 0 | ∂r 0 ≤

Secondly, for any T > 1 and r0 in the compact region [1, T + 1] we obtain for constant c = c(T ) > 0, for all s [ 1 , 1], 0 ∈ 2 ∂RH ∂2RH (s ; r ) , (s ; r ) c(T ). (5.3) ∂s 0 0 ∂s∂r 0 0 ≤

17 Integral operators If k : [0, ) R is smooth and compactly supported, which will suffice 0 ∞ → here, then one can construct a kernel

def k(x,y) = k0(dH(x,y)) with corresponding integral operator C∞(H) C∞(H) → def K[f](x) = k(x,y)f(y)dH(y) H Zy∈ where dH is the hyperbolic area form on H. Such an operator commutes with the Laplacian on H and hence preserves its generalized eigenspaces. If f C∞(H) is an generalized eigenfunction of ∆ with eigenvalue 1 + ξ2, ∈ 4 ξ 0, then by [Ber16, Thm. 3.7, Lemma 3.9] (cf. also Selberg’s original ≥ article [Sel56]) K[f]= h(ξ)f where ∞ ∞ k (ρ) sinh(ρ) h(ξ)= √2 eiξu 0 dρdu. −∞ |u| cosh(ρ) cosh(u) Z Z − By our assumptions on k0 the integralp above is convergent. It will be esti- mated more precisely for particular choice of k0 later. For now note that if k0 is real valued then h is the Fourier transform of a real valued even function and hence real-valued. Since L2(H) has a generalized eigenbasis of C∞ eigenfunctions of the Laplacian, by Borel functional calculus K extends ∞ H 2 H from e.g. Cc ( ) to a (possibly unbounded) self-adjoint operator on L ( ) with operator norm K L2(H) = sup h(ξ) . (5.4) k k ξ≥0 | |

5.2 Interior parametrix on hyperbolic plane Let χ : R [0, 1] denote a smooth function with χ 1 and 0 → 0|(−∞,0]≡ χ = 0. Then let 0|[1,∞) χ (t) def= χ (t T ) T 0 − so χ is supported on ( , T + 1]; note for later that all derivatives of χ T −∞ T are supported in [T, T + 1] and have uniform bounds independent of T . We define (T ) def RH (s; r) = χT (r)RH(s; r).

18 (T ) Let RH (s) denote the corresponding integral operator. In radial coordi- nates the Laplacian on H is given by [Ber16, (3.2)]

∂2 1 ∂ 1 ∂2 . −∂r2 − tanh r ∂r − (2 sinh r)2 ∂θ2

We now perform the following calculation, writing ∆x for the Laplacian acting on coordinate x:

(T ) [∆ s(1 s)] RH (s; r)=[∆ s(1 s)] χ (r)RH(s; r) x − − x − − T ∂2 1 ∂ = ,χ RH(s; r)+ δ (5.5) −∂r2 − tanh r ∂r T r=0   which is understood in a distributional sense. We further calculate ∂2 1 ∂ ∂2 ∂ ∂ 1 ∂ ,χ = [χ ] 2 [χ ] [χ ]. (5.6) −∂r2 − tanh r ∂r T −∂r2 T − ∂r T ∂r − tanh r ∂r T   Combining (5.5) and (5.6) we expect an identity of operators

(T ) (T ) [∆ s(1 s)] RH (s)=1+ LH (s) (5.7) − − (T ) where LH (s) is the integral operator with real-valued radial kernel

2 (T ) def ∂ 1 ∂ ∂ ∂RH LH (s; r ) = [χ ] [χ ] RH(s; r ) 2 [χ ] (s; r ). 0 −∂r2 T − tanh r ∂r T 0 − ∂r T ∂r 0   (5.8) The identity (5.7) will be established in Lemma 5.3 below. We note the following properties of this kernel that are straightforward consequences of the way χT was chosen and (5.2), (5.3). Lemma 5.1. We have 1 (T ) 1. For T > 0 and s [ , 1], LH (s; ) is smooth and supported in [T, T + ∈ 2 • 1].

2. There is a constant C > 0 such that for any T > 0 and s [ 1 , 1] we ∈ 2 have (T ) −sr0 LH (s; r ) Ce | 0 |≤ 3. For any T > 0 there is a constant c(T ) > 0 such that for any r 0 ∈ [T, T + 1] (T ) ∂LH (s0; r0) c(T ). ∂s ≤

19 We can now show the following.

Lemma 5.2. There is a constant C > 0 such that for any T > 0 and 1 (T ) 2 s [ , 1] the operator LH (s) extends to a bounded operator on L (H) with ∈ 2 operator norm

1 (T ) ( 2 −s)T LH (s) 2 CT e . k kL ≤ Proof. We do this using (5.4) which tells us

∞ ∞ L(T ) iξu k0(ρ) sinh(ρ) H (s) L2 sup √2 e dρdu k k ≤ ξ≥0 −∞ |u| cosh(ρ) cosh(u) Z Z − ∞ ∞ (T ) LH (s;pρ) sinh(ρ) √2 | | dρdu ≤ −∞ |u| cosh(ρ) cosh(u) Z Z − T +1 T +1 e−sρ sinh(ρ) 2√2C p dρdu ≤ 0 max(|u|,T ) cosh(ρ) cosh(u) Z Z − T +1 T +1 sinh(ρ) C′e−sT p dρdu ≤ 0 max(|u|,T ) cosh(ρ) cosh(u) Z Z − T +1 cosh(T +1) dy = C′e−sT p du 0 cosh max(|u|,T ) y cosh(u) Z Z − T +1 = C′′e−sT cosh(T + 1)p cosh u 0 − Z hp cosh max( u , T ) cosh u du − | | − | | 1 i C′′′T e( 2 −s)T p ≤ where the third inequality used Lemma 5.1.

1 The implication of this is that for any s0 > 2 we can choose T = T (s0) such that for all s [s , 1] we have ∈ 0 (T ) 1 LH (s) 2 . k kL ≤ 5 We will do this later. The following lemma shows that smoothly cutting off RH(s) at radius T does not significantly affect its mapping properties.

Lemma 5.3. For any T > 0 and s [ 1 , 1], for any compact K H, there ∈ 2 ⊂ is C = C(s,K,T ) > 0 such that:

20 ∞ (T ) 2 1. For any f C (H) with supp(f) K we have RH (s)f H (H) ∈ c ⊂ ∈ and (T ) RH (s)f 2 C(s,K,T ) f 2 . k kH ≤ k kL 2. Furthermore, with f as above

(T ) (∆ s(1 s))RH (s)[f]= f + LH(s)[f] − − in the sense of equivalence of L2 functions. ∞ H Proof. Suppose that compact K is given and f Cc ( ) with supp(f) K. (T ) ∈ ⊂ For y K we have RH (s; x,y) = 0 unless ∈ x K′(T, K) def= x : d(x, K) T + 1 ∈ { ≤ } with K′ compact. Therefore using the usual Hilbert-Schmidt inequality we obtain

(T ) 2 RH (s)[f] 2 H k kL ( ) 2 (T ) = RH (s; x,y)f(y)dH(y) dH(x) ′ Zx∈K Zy∈K

(T ) 2 H 2 H H RH (s; x,y) d (y) f(y) d (y) d (x). (5.9) ≤ ′ | | Zx∈K Zy∈K Zy∈K  Recall that we write r = dH(x,y), hence the inner integral can be written in polar coordinates as

2π ∞ (T ) 2 (T ) 2 RH (s; x,y) dH(y)= RH (s; r) sinh r dr dθ Zy∈K Z0 Z0 M (T ) 2 2π RH (s; r) sinh r dr (5.10) ≤ Z0 for M = M(K, T ). Because χT 1 near 0, the type of singularity that (T ) ≡ RH,n(s; r) has at r = 0 is exactly the same as the type of singularity of RH(s; r) near r = 0; namely by [Bor16, (4.2)]

(T ) 1 r RH (s; r)= log + O(1) (5.11) −4π 2   (T ) as r 0. The function RH (s; r) is smooth away from 0. Hence, since →r 2 (T ) log sinh r 0 as r 0, RH (s; r) is in particular square integrable 2 → →

21 on [0, M]. This gives from (5.9)

(T ) 2 2 H ′ 2 RH (s)[f] L2(H) C(s,K,T ) f L2(H)d (x) C (s,K,T ) f L2 (H). k k ≤ x∈K′ k k ≤ k k Z (5.12) (T ) 2 (T ) We now aim for a bound on ∆RH (s)[f] 2 so as to prove RH (s)[f] k kL (H) ∈ H2(H) and bound its H2-norm. Let g C∞(H) be a test function and f C∞(H) with support as ∈ c ∈ c above. Consider

(T ) (T ) RH (s)[f], ∆g = RH (s; x,y)f(y)dH(y) ∆g(x)dH(x). h i H H Zx∈ Zy∈  (T ) Because f and g are compactly supported and the singularity of RH (x,y) is locally L1 using (5.11), we can use Fubini to get

(T ) (T ) RH (s)[f], ∆g = RH (s; x,y)∆g(x)dH(x) f(y)dH(y). h i H H Zy∈ Zx∈  (5.13) We use hyperbolic polar coordinates for the inner integral, writing r = def d(x,y) and θ for polar angle, Gy(r, θ) =g ¯(x), and the inner integral is understood as improper integral as follows:

(T ) RH (s; x,y)∆g(x)dH(x) H Zx∈ ∞ 2π 2 (T ) ∂ ∂Gy 1 ∂ Gy = lim RH (s;r ˜) sinh r (˜r, θ˜)+ (˜r, θ˜) dθd˜ r˜ − ǫ→0 ∂r ∂r (sinh r)2 ∂θ2 Zǫ Z0     ∞ 2π (T ) ∂ ∂Gy = lim RH (s;r ˜) sinh r (˜r, θ˜) dθd˜ r˜ − ǫ→0 ∂r ∂r Zǫ Z0     2π ∞ (T ) ∂ ∂Gy = lim RH (s;r ˜) sinh r (˜r, θ˜) drd˜ θ˜ − ǫ→0 ∂r ∂r Z0 Zǫ     2π (T ) ∂Gy = lim RH (s; ǫ)sinh ǫ (ǫ, θ˜)dθ˜ ǫ→0 ∂r Z0 2π ∞ (T ) ∂RH ∂G + lim (s;r ˜) sinhr ˜ y (˜r, θ˜)drd˜ θ˜ ǫ→0 ∂r ∂r Z0 Zǫ 2π ∞ (T ) ∂RH ∂G = lim (s;r ˜) sinhr ˜ y (˜r, θ˜)drd˜ θ˜ (5.14) ǫ→0 ∂r ∂r Z0 Zǫ where the last equality used (5.11) with smoothness of Gy. Now a similar

22 calculation gives

2π ∞ (T ) ∂RH ∂G lim (s;r ˜)sinh r y (˜r, θ˜)drd˜ θ˜ ǫ→0 ∂r ∂r Z0 Zǫ (T ) 2π ∂RH = lim (s; ǫ)sinh ǫ Gy(ǫ, θ˜)dθ˜ − ǫ→0 ∂r Z0 2π ∞ (T ) ∂ ∂RH lim sinh r (s;r ˜)Gy(˜r, θ˜)drd˜ θ˜ − ǫ→0 ∂r ∂r Z0 Zǫ ! 2π ∞ (T ) ∂ ∂RH =¯g(y) lim sinh r (s;r ˜)Gy(˜r, θ˜)drd˜ θ.˜ (5.15) − ǫ→0 ∂r ∂r Z0 Zǫ ! The second equality used [Bor16, pg. 66]

(T ) ∂RH 1 (s; ǫ)= + O(1) ∂r −2πǫ as ǫ 0 with lim G (ǫ, θ˜) =g ¯(y). Now we note → ǫ→0 y (T ) 1 ∂ ∂RH (T ) sinh r (s;r ˜)=∆RH (s;r ˜) −sinh r ∂r ∂r ! and so using (5.7) and (5.8) we get

(T ) 1 ∂ ∂RH (T ) (T ) sinh r (s;r ˜)= s(1 s)RH (s;r ˜)+ LH (s;r ˜). −sinh r ∂r ∂r ! −

Therefore

2π ∞ (T ) ∂ ∂RH lim sinh r (s;r ˜)Gy(˜r, θ˜)drd˜ θ˜ − ǫ→0 ∂r ∂r Z0 Zǫ ! (T ) (T ) = lim s(1 s)RH (s;r ˜)+ LH (s; d(x,y)) g¯(x)dH(x) ǫ→0 d(x,y)>ǫ − Z   (T ) (T ) = s(1 s)RH (s; d(x,y)) + LH (s; d(x,y)) g¯(x)dH(x) (5.16) − Z   and this last integral is easily seen to converge by working in polar coordi- nates centered at y and using g C∞(H) and (5.11). ∈ c

23 Now combining (5.14), (5.15), and (5.16) gives

(T ) RH (s)[f], ∆g h i = f(y)¯g(y)dH(y) H Zy∈ (T ) (T ) + s(1 s)RH (s; d(x,y)) + LH (s; d(x,y)) g¯(x)H(x) f(y)dH(y) − Z Z   (T ) (T )  = f, g + s(1 s)RH (s)[f], g + LH (s)[f], g . h i h − i h i Note that by (5.12) and Lemma 5.2 all functions above are in L2(H). This identity now clearly extends to any g H2(H) and now self-adjointness of 2 ∈(T ) 2 ∆H on H (H) (see [Str83]) gives that RH (s)[f] H (H) and moreover ∈ (T ) (T ) (∆ s(1 s))RH (s)[f]= f + LH (s)[f] − − in the sense of elements of L2(H). This proves part 2 of the lemma. We now rewrite this identity as

(T ) (T ) (T ) ∆RH (s)[f]= f + s(1 s)RH (s)[f]+ LH (s)[f] − and using Lemma 5.2 and (5.12) now gives

(T ) ∆RH (s)[f] 2 H c(s,K,T ) f 2 H . k kL ( ) ≤ k kL ( ) Combining this with (5.12), this proves part 1 of the lemma.

5.3 Automorphic kernels At two points in the rest of the paper we use the following geometric lemma. Recall that F is a Dirichlet fundamental domain for Γ in H.

Lemma 5.4. For any compact set K F¯, and T > 0, there is another ⊂ compact set K′ = K′(K, T ) F¯ and a finite set S = S(K, T ) Γ such that ⊂ ⊂ for all x,y H with ∈ x F,¯ y γ(K), d(x,y) T, ∈ ∈ ≤ γ[∈Γ we have x K′, y γ(K). ∈ ∈ γ[∈S

24 Proof. Suppose that y = γy′ with y′ K, x F¯, γ Γ and d(x,y) T . ∈ ∈ ∈ ≤ Then d(γy′,x)= d(y′, γ−1x) T implies γ−1F intersects the compact set ≤ z : d(z, K) T . { ≤ } Since we took F to be a Dirichlet fundamental domain, by [Bea83, Thm. 9.4.2] there are only a finite number of γ for which this is possible. Let S be this finite subset of Γ. Then furthermore, if we let o denote a fixed point in K we have

d(x, o) d(x,γy′)+ d(γy′, γo)+ d(γo, o). ≤ T + d(y′, o)+ d(γo, o) C′(K, T ). ≤ ≤ This implies that x must be in some compact set K′ = K′(K, T ) F¯. ⊂ Define (T ) def (T ) 0 RH,n(s; x,y) = RH (s; x,y)IdVn , (T ) def (T ) L L 0 H,n(s; x,y) = H (s; x,y)IdVn , (T ) L(T ) and RH,n(s), H,n(s) the corresponding integral operators. In the next lemma we describe the mapping properties of these integral operators and describe a functional identity that they satisfy that arises from (5.7). In the follow- − ∞ ing we regard the function χC defined in 4 as a Γ-invariant C function H −§ 2 on . The action of multiplication by χC,φ on new functions in Lnew(Xφ) − corresponds to multiplication by the scalar-valued invariant function χC on 2 H 0 Lφ( ; Vn ). Lemma 5.5. For all s [ 1 , 1], ∈ 2 (T ) − ∞ 0 1. The integral operator RH (s)(1 χ ) is well-defined on C (H; V ) ,n − C c,φ n and extends to a bounded operator

(T ) − 2 0 2 0 RH (s)(1 χ ) : L (H; V ) H (H; V ). ,n − C φ n → φ n L(T ) − ∞ H 0 2. The integral operator H,n(s)(1 χC ) is well-defined on Cc,φ( ; Vn ) − 2 H 0 and and extends to a bounded operator on Lφ( ; Vn ). 3. We have

(T ) − − (T ) − [∆ s(1 s)] RH (s)(1 χ ) = (1 χ )+ LH (s)(1 χ ) (5.17) − − ,n − C − C ,n − C 2 H 0 as an identity of operators on Lφ( ; Vn ).

25 Proof. Suppose first that f C∞ (H; V 0) (i.e. automorphic, smooth, and ∈ c,φ n compactly supported modulo Γ). We have for x F ∈ (T ) − def (T ) − H RH,n(s)(1 χC )[f](x) = RH (s; x,y)(1 χC (y))f(y)d (y). (5.18) − H − Zy∈ The integrand here is non-zero unless d(x,y) T +1 and y is in the ≤ the support of 1 χ−, which is a union of the Γ-translates of a compact − C set K of of F¯. Applying Lemma 5.4 tells us then that there is compact K1 = K1(T ) F¯ and finite set S = S(T ) such that the integrand in (5.18) ⊂ def is supported on the compact set K = γ−1K and the whole integral is 2 ∪γ∈S zero unless x K (given x F¯ to begin with). ∈ 1 ∈ Let ψ be a smooth function that is 1 in K K, valued in [0, 1] and ≡ 2 ∪ compactly supported. Let ei : i [n 1] denote an orthonormal basis 0 { ∈ − } for Vn and let def f = f, e C∞(H). i h ii∈ The above shows that for x F we have ∈ (T ) − (T ) − RH (s)(1 χ )[f](x)= RH (s)(1 χ )[ψf](x) ,n − C ,n − C n−1 (T ) − = RH (s) 1 χ ψf (x)e (5.19) − C i i i=1 X 
 hence

n−1 2 (T ) − 2 (T ) − RH (s)(1 χ )[f](x) 0 = RH (s) 1 χ ψfi (x) , k ,n − C kVn − C Xi=1 n−1 
 2 (T ) − 2 (T ) − ∆RH (s)(1 χ )[f](x) 0 = ∆RH (s) 1 χ ψfi (x) . k ,n − C kVn − C i=1 X 
 Each function 1 χ− ψf is smooth here and has has compact support − C i depending only on T and χ−.
C Now using Lemma 5.3 Part 1, the fact that 1 χ− is valued in [0, 1], − C using ψ is supported only on finitely many Γ-translates of F , together with
automorphy of f, we get by integrating over F

(T ) − 2 2 ′ 2 RH (s)(1 χ )[f] 2 C ψf 2 H C f 2 , k ,n − C kL (F ) ≤ k ikL ( ) ≤ k kL (F ) Xi (T ) − 2 2 ′ 2 ∆RH (s)(1 χ )[f] 2 C ψf 2 H C f 2 , k ,n − C kL (F ) ≤ k ikL ( ) ≤ k kL (F ) Xi

26 where C,C′ depend on s, T . Now this bound clearly extends to f L2 (H; V 0). ∈ φ n This proves the first statement of the lemma. L(T ) 2 H 0 The statement that H,n(s) is well-defined and bounded on Lφ( ; Vn ) is just an easier version of the previous proof using Lemma 5.2 instead of Lemma 5.3. This proves the second statement of the lemma. We note that we also obtain

n−1 (T ) − (T ) − LH (s)(1 χ )[f]= LH (s) 1 χ ψf e (5.20) ,n − C − C i i i=1 X 
 analogously to (5.19). Now going back to (5.19) and using Lemma 5.3 Part 2 gives, considered by restriction as equivalence classes of measurable functions on F ,

(T ) − (∆ s(1 s))RH (s)(1 χ )[f] − − ,n − C n−1 (T ) − = (∆ s(1 s))RH (s) 1 χ ψf e − − − C i i Xi=1 n−1 
 − (T ) − = 1 χ ψf + LH (s) 1 χ ψf e − C i − C i i Xi=1   −
(T ) −
 = 1 χ f + LH (s) 1 χ [f] . − C ,n − C On the other hand, the
fact that all functions
at the two ends of the string of equalities above satisfy the automorphy equation (2.1) almost everywhere implies that indeed

(T ) − − (T ) − (∆ s(1 s))RH (s)(1 χ )[f]= 1 χ f + LH (s) 1 χ [f] − − ,n − C − C ,n − C as equivalence classes of measurable functions
on H. This proves the
final part of the lemma.

Lemma 5.5 allows us to define our interior parametrix

Mint(s) : L2 (X ) H2 (X ) φ new φ → new φ 2 2 H 0 2 to be the operator corresponding under Lnew(Xφ) ∼= Lφ( ; Vn ) and Hnew(Xφ) ∼= 2 H 0 (T ) − Hφ( ; Vn ) to the integral operator RH,n(s)(1 χC ). The equation (5.17) be- 1 − comes for s> 2

(∆ s(1 s))Mint(s) = (1 χ− )+ Lint(s) (5.21) Xφ − − φ − C,φ φ

27 where Lint(s) : L2 (X ) L2 (X ) φ new φ → new φ 2 2 H 0 is the operator corresponding under Lnew(Xφ) ∼= Lφ( ; Vn ) to the integral (T ) − operator LH (s)(1 χ ) (the dependence on T is suppressed here but re- ,n − C membered). Therefore if we define M def Mint Mcusp φ(s) = φ (s)+ φ (s), we have M (s) : L2 (X ) H2 (X ) φ new φ → new φ and combining (4.7) and (5.21) we obtain

∆ s(1 s) M (s) = (1 χ− )+ Lint(s)+ χ− + Lcusp(s) Xφ − − φ − C,φ φ C,φ φ Lint Lcusp
=1+ φ (s)+ φ (s). (5.22) The aim is to show, with high probability, we can invert the right hand 1 side of (5.22) to define a bounded resolvent in s s0 > 2 . Since we have Lcusp ≥ already suitably bounded φ (s) in Proposition 4.4, it remains to bound Lint the operator φ (s); this random operator will be studied in detail in the next section.

6 Random operators

6.1 Set up

∞ 0 2 Suppose that f C (H; V ) with f 2 < . We have ∈ φ n k kL (F ) ∞

L(T ) − L(T ) − H,n(s)(1 χC )[f](x)= H,n(s; x,y)(1 χC (y))f(y) − H − Zy∈ L(T ) 0 −1 − = H,n(s; γx,y)ρn(γ )(1 χC (y))f(y). y∈F − Xγ∈Γ Z (6.1)

(T ) Using that LH (s; x,y) localizes to d(x,y) T + 1, Lemma 5.4 implies that ,n ≤ there is a compact set K = K(T ) F and a finite set S = S(T ) Γ such ⊂ ⊂ that for x,y F¯ and γ Γ ∈ ∈ (T ) 0 −1 − LH (s; γx,y)ρ (γ )(1 χ (y)) = 0 ,n n − C

28 unless x,y K and γ Γ. ∈ ∈ The point of view we take in the rest of this section is that there is an isomorphism of Hilbert spaces

L2 (H; V 0) = L2(F ) V 0; φ n ∼ ⊗ n f f , e e 7→ h |F ii⊗ i e Xi 0 where ei is an orthonormal basis of Vn (the choice of this basis does not matter). When conjugated by this isomorphism, (6.1) shows that

(T ) − 0 −1 LH (s)(1 χ ) = a (s) ρ (γ ) ,n − C ∼ γ ⊗ n γX∈S where

a (s) : L2(F ) L2(F ) γ → def (T ) − a (s)[f](x) = LH (s; γx,y)(1 χ (y))f(y)dH(y). γ − C Zy∈F Again, this sum is supported on a finite set S = S(T ) Γ depending on T − ⊂ (and the fixed χC ). (T ) Since LH (s; γx,y) is bounded depending on T and smooth we have

(T ) 2 − 2 LH (s; γx,y) (1 χ (y)) dH(x)dH(y)= | | − C Zx,y∈F (T ) 2 − 2 LH (s; γx,y) (1 χ (y)) dH(x)dH(y) < | | − C ∞ Zx,y∈K shows that each integral operator aγ(s) is bounded, and in fact Hilbert- Schmidt, so compact. We also produce the following deviations estimate for the aγ(s).

Lemma 6.1. For fixed T > 0, there is a constant c1 = c1(T ) > 0 such that for s ,s [ 1 , 1] and γ S(T ) we have 1 2 ∈ 2 ∈ a (s ) a (s ) 2 c s s . k γ 1 − γ 2 kL (F ) ≤ 1| 1 − 2| Proof. Suppose s ,s [ 1 , 1], and γ is fixed. The operator a (s ) a (s ) 1 2 ∈ 2 γ 1 − γ 2 is a Hilbert-Schmidt operator on L2(F ) with kernel

(T ) (T ) − [LH (s ; γx,y) LH (s ; γx,y)](1 χ (y)); 1 − 2 − C

29 once again this is zero unless x,y are in compact set K(T ). We have by Lemma 5.1 Part 3 that for all x,y K ∈ (T ) (T ) LH (s ; γx,y) LH (s ; γx,y) c(T ) s s . 1 − 2 ≤ | 1 − 2|

It follows that there is a constant c = c (T ) such that, writing for 1 1 k • kH.S. Hilbert-Schmidt norm of a Hilbert-Schmidt operator on L2(F ), we have

(T ) (T ) − a (s ) a (s ) 2 [LH (s ) LH (s )](1 χ ) k γ 1 − γ 2 kL (F ) ≤ k 1 − 2 − C kH.S. c (T ) s s . ≤ 1 | 1 − 2| Finally, because γ is in the finite set S(T ), the estimate can be taken uni- formly over all γ S(T ) by taking the maximal value of c . ∈ 1 6.2 The operator on the free group We are momentarily going to apply Corollary 3.4 to the random operator

(T ) def= a(T )(s) ρ0 (γ−1) Ls,φ γ ⊗ n γX∈S Up to an intermediate step, where we approximate the coefficients aγ (s) by finite rank operators, we expect to learn that the operator norm of (T ) is Ls,φ close to the operator norm of

def (T ) = a(T )(s) ρ (γ−1) Ls,∞ γ ⊗ ∞ γX∈S on L2(F ) ℓ2(Γ), where ⊗ ρ :Γ End(ℓ2(Γ))) ∞ → is the right regular representation of Γ. So we must understand the operator (T ) s,∞. Under the isomorphism L L2(F ) ℓ2(Γ) = L2(H), ⊗ ∼ f δ f γ−1 ⊗ γ 7→ ◦ −1 (T ) (with f γ extended by zero from a function on γF ) the operator s,∞ is ◦ L conjugated to none other than

(T ) − 2 2 LH (s)(1 χ ) : L (H) L (H) − C → from 5.2. Since (1 χ−) is valued in [0, 1], multiplication by it has operator §§ − C norm 1 on L2(H). Hence we obtain the following corollary of Lemma 5.2. ≤

30 1 Corollary 6.2. For any s0 > 2 there is T = T (s0) > 0 such that for any s [s , 1] we have ∈ 0 (T ) 1 2 2 . kLs,∞kL (F )⊗ℓ (Γ) ≤ 5 6.3 Probabilistic bounds for operator norms The aim of this section is to prove the following result as a consequence of Corollary 3.4. 1 Proposition 6.3. For any s0 > 2 there is T = T (s0) > 0 such that for s fixed in [s , 1], with probability tending to one as n , 0 →∞ (T ) 2 2 0 . kLs,φ kL (F )⊗Vn ≤ 5 Proof. Let T be the one provided by Corollary 6.2 for the given s0, so that

(T ) 1 2 2 . (6.2) kLs,∞kL (F )⊗ℓ (Γ) ≤ 5 Fix s [s , 1]. Recall that the coefficients a (s) are supported on a finite set ∈ 0 γ S = S(T ) Γ. Because the a (s) are Hilbert-Schmidt, and hence compact, ⊂ γ we can find a finite-dimensional subspace W L2(F ) ⊂ (T ) and operators bγ (s) : W W such that → (T ) (T ) 1 b (s) a (s) 2 k γ − γ kL (F ) ≤ 20 S(T ) | | for all γ S(T ). Therefore, since each ρ0 (γ) is unitary on V 0 we get ∈ n n (T ) (T ) 0 −1 1 b (s) ρ (γ ) 2 0 . (6.3) kLs,φ − γ ⊗ n kL (F )⊗Vn ≤ 20 γX∈S (T ) 0 −1 We now apply Corollary 3.4 to the operator bγ (s) ρ (γ ) to γ∈S ⊗ n obtain that with probability 1 as n , → →∞ P (T ) 0 −1 (T ) −1 1 b (s) ρ (γ ) 0 b (s) ρ (γ ) 2 + . (6.4) k γ ⊗ n kW ⊗Vn ≤ k γ ⊗ ∞ kW ⊗ℓ (Γ) 20 γX∈S γX∈S 1 (Above, 20 could be replaced by any ǫ> 0 but this is not needed here.) On the other hand, we also have by the same argument as led to (6.3)

(T ) (T ) −1 1 b (s) ρ (γ ) 2 . (6.5) kLs,∞ − γ ⊗ ∞ kL(F )⊗ℓ (Γ) ≤ 20 γX∈S Then combining (6.2), (6.3), (6.4), and (6.5) gives the result.

31 7 Proof of Theorem 1.1

Given ǫ> 0 let s = 1 + √ǫ so that s (1 s )= 1 ǫ. We assume χ± were 0 2 0 − 0 4 − C chosen as in Proposition 4.4 for this s0 so that

cusp 1 L (s) 2 (7.1) k φ kL ≤ 5 for all s (s , 1]. ∈ 0 Let T = T (s0) be the value provided by Proposition 6.3 for this s0. To control all values of s [s , 1] we use a finite net. For s ,s [s , 1] we have ∈ 0 1 2 ∈ 0 (T ) (T ) = [a(T )(s ) a(T )(s )] ρ0 (γ−1) (7.2) Ls1,φ −Ls2,φ γ 1 − γ 2 ⊗ n γX∈S where S Γ is a finite set depending on T . Lemma 6.1 tells us that for ⊂ c1 = c1(T ) > 0 we have

a (s ) a (s ) 2 c s s k γ 1 − γ 2 kL (F ) ≤ 1| 1 − 2| for all γ S and s1,s2 [s0, 1]; hence we obtain from (7.2) and as each 0 −1 ∈ ∈ ρn(γ ) is unitary that

(T ) (T ) S c s s . (7.3) kLs1,φ −Ls2,φk ≤ | | 1| 1 − 2|

We choose a finite set Y , depending on s0, of points in [s0, 1] such that every point of [s0, 1] is within 1 5 S c | | 1 of some element of Y . Now, applying Proposition 6.3 to each point of Y we obtain that with probability tending to one as n we have →∞ (T ) 2 2 0 kLs,φ kL (F )⊗Vn ≤ 5 for all s Y . Hence also using (7.3) we obtain that with probability tending ∈ to one as n we have →∞ (T ) 3 2 0 kLs,φ kL (F )⊗Vn ≤ 5 for all s [s , 1]. ∈ 0 Recall that we defined M def Mint Mcusp φ(s) = φ (s)+ φ (s).

32 1 M By Lemma 5.5 and the discussion after (4.6), for s> 2 φ(s) is a bounded 2 2 operator from Lnew(Xφ) to Hnew(Xφ). We also proved in (5.22) ∆ s(1 s) M (s)=1+ Lint(s)+ Lcusp(s) Xφ − − φ φ φ
(T ) on L2 (X ). The operator Lint(s) is unitarily conjugate to as in new φ φ Ls,φ 6.1 and hence a.a.s. has operator norm at most 3 . Hence also using the §§ 5 deterministic bound (7.1) we obtain that

int cusp 4 L (s)+ L (s) 2 k φ φ kLnew(Xφ) ≤ 5 and hence a.a.s. −1 Lint Lcusp 1+ φ (s)+ φ (s)  2  exists as a bounded operator from Lnew(Xφ) to itself. We now get −1 ∆ s(1 s) M (s) 1+ Lint(s)+ Lcusp(s) = 1 Xφ − − φ φ φ   which shows there is a bounded
right inverse to ∆ s(1 s) ; this shows Xφ − − that ∆ s(1 s) maps H2 (X ) onto L2 (X ) and since it is self- Xφ − − new φ new φ
adjoint for s [ 1 , 1], cannot have any kernel in L2 (X ). This shows a.a.s. ∈ 2
new φ that ∆ has no new eigenvalues λ with λ s (1 s ) = 1 ǫ. This Xφ ≤ 0 − 0 4 − concludes the proof of Theorem 1.1.

8 Proof of Corollary 1.3

Take X to be a once-punctured torus or thrice-punctured sphere and apply Corollary 1.2 to obtain a sequence of connected orientable finite area non- compact hyperbolic surfaces X with χ(X )= i and i i − 1 inf (spec(∆ ) (0, )) . Xi ∩ ∞ → 4

Suppose Xi has genus gi and ni cusps. We have i = χ(X ) = 2 2g n − i − i − i which shows i χ(X ) n mod 2. ≡ i ≡ i In particular, for even i, we have an even number of cusps in Xi. We use the following result of Buser, Burger, and Dodziuk [BBD88]. See Brooks and Makover [BM01, Lemma 1.1] for the extraction of this lemma from [BBD88].

33 Lemma 8.1 (Handle Lemma). Let X be a finite area hyperbolic surface with an even number of cusps C . It is possible to deform the surface { i} X in a certain way to a finite area hyperbolic surface with boundary, where each cusp becomes a bounding geodesic of length t, and then glue the geodesic corresponding to C2i−1 to the one corresponding to C2i to form a family of closed hyperbolic surfaces X(t) such that

(t) lim sup λ1(X ) inf (spec(∆X ) (0, )) . t→0 ≥ ∩ ∞

In particular, since each X2k has an even number of cusps, we can con- struct a closed hyperbolic surface X˜ of Euler characteristic 2k and 2k − 1 λ (X˜ ) inf (spec(∆ ) (0, )) . 1 2k ≥ X2k ∩ ∞ − k

On the other hand, the upper bound of Huber [Hub74] now implies λ1(X˜2k) 1  → 4 as required.

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Will Hide, Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, United Kingdom [email protected]

Michael Magee, Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, United Kingdom [email protected]

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