Counting saddle connections in a homology class modulo $q$

Magee, Michael; Rühr, Rene; Gutiérrez-Romo, Rodolfo

doi:10.48550/arxiv.1809.00579

Cited by 1 publication

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References 18 publications

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“…We mention that in recent work [MR18], joint with Rühr and Gutiérrez-Romo, we extend Theorem 1.4 to congruence covers coming from relative homology of (S, Σ), and apply both Theorem 1.4 and the extended result to the problem of counting saddle connections in a homology class modulo q.…”

Section: Introductionmentioning

confidence: 99%

On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

Magee¹

2019

Compositio Math.

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J.-C. Yoccoz proposed a natural extension of Selberg's Eigenvalue Conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg's 3 16 Theorem to moduli spaces of abelian differentials on surfaces of genus ≥ 2.• There is an action of SL 2 (R) on M. The restriction of the SL 2 (R) action to the one parameter diagonal subgroup gives a flow on M called the Teichmüller flow that generalizes the geodesic flow on the unit tangent bundle of X.• There is a unique probability measure ν M on M that is SL 2 (R)-invariant, ergodic for the Teichmüller flow, and in the Lebesgue class with respect to a natural affine orbifold structure on M. This is due to works of Masur [Mas82] and Veech [Vee82].• The space SO(2)\M is locally foliated by H and hence it is possible to define a foliated Laplacian ∆ M on SO(2)\M generalizing ∆ X . This operator has a simple eigenvalue at zero and by a result of Avila and Gouëzel [AG13], its spectrum below 1 4 has no accumulation points other than possibly at 1 4 . Each of these objects lifts to M(q), so there is an SL 2 (R) action on M(q) preserving a finite measure ν M(q) , and a foliated Laplacian ∆ M(q) whose spectrum below 1 4 does not accumulate 2 away from 1 4 . Hence we can write λ 1 (M(q)) for the infimum of the non-zero spectrum 3 of ∆ M(q) . The following extension of Selberg's conjecture to genus g ≥ 2 was proposed by Yoccoz 4 . Conjecture 1.3 (Yoccoz). For all q ≥ 2, and any connected component M of a stratum, A. λ 1 (M(q)) ≥ 1 4 .B. The measure on the unitary dual of SL 2 (R) that decomposes L 2 (M(q), ν M(q) ) is supported away from complementary series representations.The main theorem of this paper gives an approximation to Conjecture 1.3.Theorem 1.4. For any connected component M of a stratum, there exists ǫ, η > 0 and Q 0 ∈ Z + such that for all q coprime to Q 0 the following hold.B. The measure on the unitary dual of SL 2 (R) that decomposes L 2 (M(q), ν M(q) ) is supported away from complementary series representations Comp u with u ∈ (1 − η, 1).C. The Teichmüller flow on M(q) has exponential decay of correlations on compactly supported C 1 observables with a rate of decay that is independent of q.2 By [AG13, Remark 2.4] this result also applies to M(q).3 In contrast to the situation with X, where it is known [Sel56] that there are infinitely many eigenvalues of ∆X , we do not know whether ∆M or ∆ M(q) have any non-zero eigenvalues. 4 The formulation of the conjecture appears in print in [AG13], although Avila and Gouëzel stopped short of making the conjecture because of lack of evidence. We learned from C. Matheus that Yoccoz had made this conjecture in private.

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Section: Introductionmentioning

confidence: 99%