A Spectral Gap for Spinors on Hyperbolic Surfaces

The purpose of this note is to construct a sequence of spin hyperbolic surfaces ${\mathrm{\Sigma }}_n$ with genus going to infinity and with a uniform spectral gap for the Dirac operator. Our construction is completely explicit. In particular, the ${\mathrm{\Sigma }}_n$ can be taken to be a tower of covers, with each ${\mathrm{\Sigma }}_n$ an arithmetic hyperbolic surface.