## 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
- J. Amer. Math. Soc.
**29**(2016), 1069-1115 - DOI: https://doi.org/10.1090/jams/849
- Published electronically: November 2, 2015
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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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## Bibliographic 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.
- © 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

Dedicated: Dedicated to Peter Sarnak on the occasion of his sixty-first birthday.