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

Abstract

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.

Counting saddle connections in a homology class modulo <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol q $\end{document}</tex-math></inline-formula> (with an appendix by Rodolfo Gutiérrez-Romo)

“…Establishing this uniform spectral gap for the covering spaces arising from nonclosed saddle connections is a main point of this paper. We extend the results of [17] to congruence covers of M that arise from relative homology. This extended result is given in Theorem 2.1.…”

“…For a family of q, there is a uniform spectral gap if η can be taken to be the same for all q. A uniform spectral gap result for the M (Θ σ q ) that arise, in the current context, from configurations of closed saddle connections, was obtained by Magee in [17]. Establishing this uniform spectral gap for the covering spaces arising from nonclosed saddle connections is a main point of this paper.…”

“…Theorem 2.1 was proved when σ = 0 (i.e., z 1 = z 2 ) by Magee in [17], using the work of Gutiérrez-Romo [12, Theorem 1.1] as a crucial input. Throughout the rest of Section 2 we therefore assume σ = z 1 − z 2 = 0.…”

“…Before discussing the spectral gap, we explain what it means. There are several different ways to state the result (see [17,Introduction] for a discussion), but the one we will use here is in terms of representation theory. There is a family of irreducible unitary representations of SL 2 (R) called complementary series representations.…”

“…We do this by establishing two facts that can be inserted into the framework of [17] as 'black boxes' to obtain the extension. The framework of [17] uses Veech's zippered rectangles construction. This associates to each M a collection of Rauzy-Veech monoids.…”