Strong spectral gap for geometrically finite hyperbolic manifolds

TL;DR

This paper proves a strong spectral gap for certain geometrically finite hyperbolic manifolds, confirming a conjecture and enabling explicit decay and mixing rate estimates based on the spectral gap size.

Contribution

It establishes the existence of a strong spectral gap for $L^2$ spaces on geometrically finite hyperbolic manifolds, extending previous spherical spectral gap results.

Findings

Confirmed a conjecture of Mohammadi and Oh.

Derived explicit decay rates of matrix coefficients.

Established exponential mixing of the frame flow.

Abstract

Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral gap, confirming a conjecture of Mohammadi and Oh. This extends the spherical spectral gap on $L^2(\Gamma\backslash \mathbb{H}^{d+1}) \cong L^2(\Gamma\backslash G/\operatorname{SO}(d+1))$, 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.