Spectral gap with polynomial rate for random covering surfaces

TL;DR

This paper demonstrates that random covers of hyperbolic surfaces have a spectral gap with a polynomial rate, improving understanding of eigenvalue distributions in geometric analysis.

Contribution

It applies recent theoretical work to establish an optimal polynomial rate for the spectral gap in random hyperbolic surface covers.

Findings

Spectral gap with polynomial error rate established

No new eigenvalues below a certain threshold with high probability

Results hold for large degree covers as n approaches infinity

Abstract

In this note we show that the recent work of Magee, Puder and van Handel [MPvH25] can be applied to obtain an optimal spectral gap result with polynomial error rate for uniformly random covers of closed hyperbolic surfaces. Let $X$ be a closed hyperbolic surface. We show there exists $b,c>0$ such that a uniformly random degree-$n$ cover $X_{n}$ of $X$ has no new Laplacian eigenvalues below $\frac{1}{4}-cn^{-b}$ with probability tending to $1$ as $n\to\infty$.