Spectral gap for Pollicott-Ruelle resonances on random coverings of Anosov surfaces

TL;DR

This paper proves the existence of a spectral gap for Pollicott-Ruelle resonances on large random coverings of Anosov surfaces, combining recent convergence results with analysis of the spherical mean operator.

Contribution

It establishes a spectral gap for resonances on random coverings of Anosov surfaces, extending understanding of spectral properties in geometric dynamical systems.

Findings

Spectral gap exists for large degree random coverings.

The result is expected to be optimal.

Combines convergence results with spherical mean analysis.

Abstract

Let $(M,g)$ be a closed Riemannian surface with Anosov geodesic flow. We prove the existence of a spectral gap for Pollicott--Ruelle resonances on random finite coverings of $M$ in the limit of large degree, which is expected to be optimal. The proof combines the recent strong convergence results of Magee, Puder and van Handel for permutation representations of surface groups with an analysis of the spherical mean operator on the universal cover of $M$.