Spectral gaps of random finite-area hyperbolic surfaces

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The goal of this project is to investigate the spectral gap between the bottom of the spectrum of the Laplace-Beltrami operator and the rest of the spectrum for finite-area, non-compact hyperbolic surfaces.One model of random surfaces studied will be the random covering model. One fixes a hyperbolic surface, and for each n, picks a degree n cover of the surface uniformly at random. The project seeks to determine whether as n tends to infinity, the probability that a random degree n cover of a fixed surface has an almost optimal relative spectral gap tends to 1. The spectral gap discussed must be a relative one, since the eigenvalues of the original fixed surface are also eigenvalues of any cover.Recent progress on this problem for compact surfaces has been made by Magee, Naud, and Puder. The rough plan is to follow the approach of (ibid.), however, the finite-area non-compact setting poses two new significant challenges:1. The Selberg trace formula is significantly more involved due to contributions from the continuous spectrum.2. There is no uniform comparison between word length in the fundamental group and the corresponding geodesic length in the surface, due to the presence of cusps in the surface.Overcoming these challenges will be a first focus of the project.One possible consequence of this approach is to resolve an open problem whether there exist a sequence of connected hyperbolic surfaces with genera tending to infinity and with smallest non-zero element of their spectrum tending to 1/4. Any alternative approach to this problem, particularly for finite-area non-compact surfaces, is within the scope of this project.Other directions that the project can take is the study of alternative random models of non-compact surfaces, such as Weil-Petersson model and/or combinatorial model introduced by Brooks-Makover. It is also of interest to produce hyperbolic surfaces with `spectral gaps' other than at 0; for finite-area non-compact hyperbolic surfaces the only true spectral gaps can be between 0 and 1/4. On the other hand, these surfaces have an interesting phenomenon of embedded eigenvalues that can be studied for random covers.