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Strong spectral gap for geometrically finite hyperbolic manifolds

Dubi Kelmer, Osama Khalil, Pratyush Sarkar

Abstract

Let $Γ< G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(Γ\backslash G)$ admits a strong spectral gap, confirming a conjecture of Mohammadi and Oh. This extends the spherical spectral gap on $L^2(Γ\backslash \mathbb{H}^{d+1}) \cong L^2(Γ\backslash G/\operatorname{SO}(d+1))$, which follows by the works of Lax-Phillips, Patterson, and Sullivan by different methods. As a consequence, we establish rates of decay of matrix coefficients, and of exponential mixing of the frame flow, that are explicitly determined by the size of the strong spectral gap.

Strong spectral gap for geometrically finite hyperbolic manifolds

Abstract

Let for be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than . We show that admits a strong spectral gap, confirming a conjecture of Mohammadi and Oh. This extends the spherical spectral gap on , which follows by the works of Lax-Phillips, Patterson, and Sullivan by different methods. As a consequence, we establish rates of decay of matrix coefficients, and of exponential mixing of the frame flow, that are explicitly determined by the size of the strong spectral gap.
Paper Structure (20 sections, 31 theorems, 135 equations)

This paper contains 20 sections, 31 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper and outline of the proof
  3. Relation to prior works
  4. Representation theoretic preliminaries
  5. Lie theory of G
  6. Unitary representation theory: generalities
  7. Sobolev norms
  8. Unitary representation theory of SO(n)
  9. Case $2 \mid n$
  10. Case $2 \nmid n$
  11. Unitary representation theory of G
  12. Asymptotic expansion of matrix coefficients in terms of the Harish-Chandra C-function and Laplace transforms
  13. Intertwining operators, Eisenstein integrals, and the Harish-Chandra C-function
  14. Asymptotic expansion of matrix coefficients following Edwards--Oh
  15. Non-vanishing of main terms of matrix coefficients
  16. ...and 5 more sections

Key Result

Theorem 1.3

If $\Gamma < G$ is a Zariski-dense, geometrically finite, discrete subgroup with $\delta_\Gamma > d/2$, then $L^2(\Gamma\backslash G)$ admits a strong spectral gap.

Theorems & Definitions (63)

  • Definition 1.1: Strong Spectral Gap
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5: Strong Spectral Gap Parameter
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 53 more