Spectral gap with polynomial rate for random covering surfaces

Will Hide; Davide Macera; Joe Thomas

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arXiv · 2505.08479

Spectral gap with polynomial rate for random covering surfaces

Will Hide·Davide Macera·Joe Thomas

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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 $n\to\infty$.

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BibTeX RIS

Will Hide, Davide Macera, Joe Thomas. 2025-05-13. Spectral gap with polynomial rate for random covering surfaces. https://arxiv.org/abs/2505.08479

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