Spectral gap of random covers of negatively curved noncompact surfaces

TL;DR

This paper proves that for large random covers of negatively curved surfaces, the spectral gap remains stable with high probability, extending previous results to more general curvature conditions.

Contribution

It extends the spectral gap stability result to geometrically finite surfaces with pinched negative curvature, covering more general metric conditions.

Findings

Spectral gap persists in large random covers with high probability.

Eigenvalues below the base spectrum are almost surely absent in large covers.

Results generalize previous work to pinched negative curvature settings.

Abstract

Let $(X,g)$ be a complete noncompact geometrically finite surface with pinched negative curvature $-b^2\leq K_g \leq -1$. Let $\lambda_0(\widetilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover $\widetilde{X}$. We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below $\lambda_0(\widetilde{X})-\varepsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\to \infty$. This extends a result of Hide--Magee to metrics of pinched negative curvature.