Geodesic
Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of 𝑆𝐿₂(ℤ)
Oh, Hee1 ; Winter, Dale2
1 Mathematics Department, Yale University, New Haven, Connecticut 06511 and Korea Institute for Advanced Study, Seoul, Korea
2 Department of Mathematics, Brown University, Providence, Rhode Island 02906
Journal of the American Mathematical Society, Tome 29 (2016) no. 4, pp. 1069-1115

Voir la notice de l'article provenant de la source American Mathematical Society

Let $\Gamma \operatorname {SL}_2(\mathbb {Z})$ be a non-elementary finitely generated subgroup and let $\Gamma (q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb N$. We obtain an asymptotic formula for the matrix coefficients of $L^2(\Gamma (q) \backslash \operatorname {SL}_2(\mathbb {R}))$ with a uniform exponential error term for all square free $q$ with no small prime divisors. As an application we establish a uniform resonance free half plane for the resolvent of the Laplacian on $\Gamma (q)\backslash \mathbb {H}^2$ over $q$ as above. Our approach is to extend Dolgopyat’s dynamical proof of exponential mixing of the geodesic flow uniformly over congruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingredients is the expander theory due to Bourgain-Gamburd-Sarnak.
DOI : 10.1090/jams/849

Oh, Hee 1 ; Winter, Dale 2

1 Mathematics Department, Yale University, New Haven, Connecticut 06511 and Korea Institute for Advanced Study, Seoul, Korea
2 Department of Mathematics, Brown University, Providence, Rhode Island 02906 
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Oh, Hee; Winter, Dale. Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of 𝑆𝐿₂(ℤ). Journal of the American Mathematical Society, Tome 29 (2016) no. 4, pp. 1069-1115. doi: 10.1090/jams/849

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