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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of $\operatorname {SL}_2(\mathbb {Z})$
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by Hee Oh and Dale Winter PDF
J. Amer. Math. Soc. 29 (2016), 1069-1115 Request permission

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
  • MR Author ID: 615083
  • Email: hee.oh@yale.edu
  • Dale Winter
  • Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02906
  • Email: dale_winter@brown.edu
  • 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.
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3522610