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arXiv · 2603.20552

Strong spectral gap for geometrically finite hyperbolic manifolds

Dubi Kelmer·Osama Khalil·Pratyush Sarkar
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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.

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Dubi Kelmer, Osama Khalil, Pratyush Sarkar. 2026-03-20. Strong spectral gap for geometrically finite hyperbolic manifolds. https://arxiv.org/abs/2603.20552

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