Near optimal spectral gaps for hyperbolic surfaces

By Will Hide, Michael Magee. Annals of Mathematics, 2021

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We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $\epsilon>0$, with probability tending to one as $n\to\infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in $[0,\frac{1}{4}-\epsilon)$ other than those of $X$, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $\frac{1}{4}$.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1f431a39c7230a438f3490a78ffd5a0324a577a7