Spectral gaps for noncompact hyperbolic surfaces with linearly many cusps

Abstract

We construct complete finite-area noncompact hyperbolic surfaces with linearly many cusps and a uniform spectral gap. More precisely, for every \(θ>0\), we construct a sequence \(S_{g,n(g)}\in\mathcal{M}_{g,n(g)}\) such that \(\lim\limits_{g\to\infty}\frac{n(g)}{g}=θ\) and the spectrum of the Laplacian has a uniform gap above zero. The construction is based on explicit expanding \((1,3)\)-graphs, viewed as combinatorial skeletons for pants decompositions. We also establish a Steklov-type upper bound showing that expansion cannot persist when the number of boundary vertices is much larger than the genus.