Spectral gaps of double covers of hyperbolic surfaces

Abstract

We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.

arXiv comment15 pages, 2 figures