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Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $ \operatorname{SL}_2(\mathbb{Z})$


Authors: Hee Oh and Dale Winter
Journal: J. Amer. Math. Soc. 29 (2016), 1069-1115
MSC (2010): Primary 37D35, 22E40, 37A25, 37D40, 11F72; Secondary 37F30, 11N45
DOI: https://doi.org/10.1090/jams/849
Published electronically: November 2, 2015
MathSciNet review: 3522610
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Abstract: 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.


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Additional Information

Hee Oh
Affiliation: Mathematics Department, Yale University, New Haven, Connecticut 06511 and Korea Institute for Advanced Study, Seoul, Korea
Email: hee.oh@yale.edu

Dale Winter
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02906
Email: dale_winter@brown.edu

DOI: https://doi.org/10.1090/jams/849
Received by editor(s): April 24, 2015
Received by editor(s) in revised form: August 26, 2015
Published electronically: November 2, 2015
Additional Notes: The first author was supported in part by NSF Grant 1361673.
Dedicated: Dedicated to Peter Sarnak on the occasion of his sixty-first birthday.
Article copyright: © Copyright 2015 American Mathematical Society