Extension of Alon's and Friedman's conjectures to Schottky surfaces

Abstract

Let $X=\Lambda \backslash \mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $\delta$ denote the Hausdorff dimension of the limit set of $\Lambda$. We prove that for any compact subset $\mathcal{K}\subset \{s:\Re (s)>\frac{\delta }{2}\}$, if one picks a random degree $n$ cover $X_n$ of $X$ uniformly at random, then with probability tending to one as $n\to \infty$, there are no resonances of $X_n$ in $\mathcal{K}$ other than those already belonging to $X$ (and with the same multiplicity). This result is conjectured to be the optimal one for bounded frequency resonances and is analogous to both Alon's and Friedman's conjectures for random graphs, which are now theorems due to Friedman and Bordenave-Collins, respectively.

arXiv comment34 pages, this version: fixed typos and minor change to Introduction

References in corpus (6)

• Spectral gaps without the pressure condition
• Expansion of Random Graphs: New Proofs, New Results
• Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$
• Short geodesic loops and $L^p$ norms of eigenfunctions on large genus random surfaces
• The Asymptotic Statistics of Random Covering Surfaces
• Large covers and sharp resonances of hyperbolic surfaces