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Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders

Louis Soares

Abstract

We prove that every family of coverings of any infinite-area, convex cocompact hyperbolic surface has uniform spectral gap, provided that the associated Schreier graphs form a family of two-sided expanders. This extends the results of Brooks, Burger, and Bourgain-Gamburd-Sarnak to a setting where the Laplacian has no $L^2$-eigenvalues. In particular, the notion of spectral gap needs to be redefined in terms of the resonances of the Laplacian. As an immediate corollary, we obtain uniform spectral gap for congruence covers of convex cocompact surfaces, a result previously established by Oh-Winter and Bourgain-Kontorovich-Magee. Moreover, given any convex cocompact hyperbolic surface $X$, we provide a new "universal" resonance-free region for $X$, by which we mean a region in the complex plane that contains no resonances for any finite cover of $X$. This enlarges the universal resonance-free region given by Magee-Naud. Our methods rely on the thermodynamic formalism for twisted Selberg zeta functions.

Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders

Abstract

We prove that every family of coverings of any infinite-area, convex cocompact hyperbolic surface has uniform spectral gap, provided that the associated Schreier graphs form a family of two-sided expanders. This extends the results of Brooks, Burger, and Bourgain-Gamburd-Sarnak to a setting where the Laplacian has no -eigenvalues. In particular, the notion of spectral gap needs to be redefined in terms of the resonances of the Laplacian. As an immediate corollary, we obtain uniform spectral gap for congruence covers of convex cocompact surfaces, a result previously established by Oh-Winter and Bourgain-Kontorovich-Magee. Moreover, given any convex cocompact hyperbolic surface , we provide a new "universal" resonance-free region for , by which we mean a region in the complex plane that contains no resonances for any finite cover of . This enlarges the universal resonance-free region given by Magee-Naud. Our methods rely on the thermodynamic formalism for twisted Selberg zeta functions.

Paper Structure

This paper contains 30 sections, 26 theorems, 268 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral gap and expanders
  3. Statement of results
  4. Congruence subgroups
  5. Further explicit constructions
  6. Outline of the proofs
  7. Outline of proof of Theorem \ref{['thm:dist_zero_free']}
  8. Outline of proof of main theorem
  9. Structure of the paper
  10. Notation
  11. Acknowledgments
  12. Preliminaries
  13. Hyperbolic geometry
  14. Twisted Selberg zeta functions
  15. Finite covers and Venkov--Zograf formula
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let notations and assumptions be as above. Suppose the graphs $\mathcal{G}_n$ form a family of two-sided $\epsilon$-expanders for some $\epsilon > 0$. Then there exists some $\eta = \eta(\epsilon,\Gamma) > 0$ such that Moreover, we may take $\eta = \exp(-c\epsilon^{-1})$ for some $c=c(\Gamma)>0.$

Figures (2)

  • Figure 1: Gray area: universal resonance-free region $\mathcal{U}$ of Theorem \ref{['thm:dist_zero_free']}
  • Figure 2: A configuration of Schottky disks and isometries with $m=3$

Theorems & Definitions (42)

  • Theorem 1.1: Main Theorem
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8: Universal resonance-free region
  • Corollary 1.9
  • Theorem 1.10: Resonance-free strip for $X(q)$
  • ...and 32 more