Uniform congruence counting for Schottky semigroups in SL<sub>2</sub>(𝐙)

Magee, Michael; Oh, Hee; Winter, Dale

doi:10.1515/crelle-2016-0072

Cited by 30 publications

(43 citation statements)

References 16 publications

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“…The key idea of the proof, due to Dolgopyat, is to systematically exploit oscillations of the roof function . As illustrated in the work of Oh and Winter [OW16] and Magee, Oh and Winter [MOW19], Dolgopyat’s argument works for skew transfer operators, provided the twisting unitary cocycle is constant on cylinders of length 1. The reason is that because the cocycle is locally constant, it should not interfere with the oscillations of during the argument, which is what is being exploited.…”

Section: The Dolgopyat Argument For Twisted Transfer Operatorsmentioning

confidence: 99%

“…The harder case is the alternative one, wherein we must extend the arguments of Magee, Oh, and Winter [MOW19, Proof of Lemma 29] to higher dimensions.…”

Section: The Dolgopyat Argument For Twisted Transfer Operatorsmentioning

confidence: 99%

“…The technique for carrying out the necessary high frequency estimates are due to Dolgopyat [Dol98] and extended to the current setting, with no -aspect, by Avila, Gouëzel and Yoccoz [AGY06]. The use of the Dolgopyat argument to establish -uniform versions of the high frequency estimates was first done by Oh and Winter [OW16], and then in a different setting by Magee, Oh, and Winter [MOW19]. In § 4 we explain how to extend the arguments of Avila, Gouëzel and Yoccoz in this regime to skew transfer operators.…”

Section: Introductionmentioning

confidence: 99%

“…The philosophy here, mirroring the Brooks–Burger philosophy, is that an iterate of the transfer operator looks somewhat like an iterate of the adjacency operator of a Cayley graph of . In work of Bourgain, Kontorovich, and Magee [MOW19, Appendix], an improvement was made to this method that allows one to use uniform expansion of Cayley graphs (in the current setting, furnished by property (T)) as a ‘black box’ 9 to prove -uniform estimates for transfer operators.…”

Section: Introductionmentioning

confidence: 99%

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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: The Dolgopyat Argument For Twisted Transfer Operatorsmentioning

confidence: 99%

“…The harder case is the alternative one, wherein we must extend the arguments of Magee, Oh, and Winter [MOW19, Proof of Lemma 29] to higher dimensions.…”

Section: The Dolgopyat Argument For Twisted Transfer Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Magee¹

2019

Compositio Math.

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“…Added in print: A power savings error now is known, and even for all q (not just square-free) by work of Magee-Oh-Winter/Bourgain-Kontorovich-Magee[MOW16,BKM15]. For our purposes, the weaker result in[BGS11] suffices, and in fact none of our estimates would improve (though the exposition would be slightly simpler) if we used[MOW16,BKM15] instead.…”

mentioning

confidence: 98%

Beyond Expansion IV: Traces of Thin Semigroups

Bourgain

Kontorovich²

2018

Discrete Anal

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This paper constitutes Part IV in our study of particular instances of the Affine Sieve, producing levels of distribution beyond those attainable from expansion alone. Motivated by McMullen's Arithmetic Chaos Conjecture regarding low-lying closed geodesics on the modular surface defined over a given number field, we study the set of traces for certain sub-semi-groups of SL 2 (Z) corresponding to absolutely Diophantine numbers (see §1.2). In particular, we are concerned with the level of distribution for this set. While the standard Affine Sieve procedure, combined with Bourgain-Gamburd-Sarnak's resonance-free region for the resolvent of a "congruence" transfer operator, produces some exponent of distribution α > 0, we are able to produce the exponent α = 1/3 − ε. This recovers unconditionally the same exponent as what one would obtain under a Ramanujan-type conjecture for thin groups. A key ingredient, of independent interest, is a bound on the additive energy of SL 2 (Z).

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Explicit spectral gaps for random covers of Riemann surfaces

Magee

Naud

2020

Publ.math.IHES

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

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Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Cited by 30 publications

References 16 publications

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

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

Beyond Expansion IV: Traces of Thin Semigroups

Explicit spectral gaps for random covers of Riemann surfaces

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Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Cited by 30 publications

References 16 publications

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

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

Beyond Expansion IV: Traces of Thin Semigroups

Explicit spectral gaps for random covers of Riemann surfaces

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Uniform congruence counting for Schottky semigroups in SL₂(𝐙)