Geometrically infinite surfaces with discrete length spectra

Abstract

It is well-known that the length spectrum of a geometrically finite hyperbolic manifold is discrete. In this paper, we begin a study of the length spectrum for geometrically infinite hyperbolic surfaces. In this generality, it is possible that the spectrum is not discrete and the main focus of this work is to find necessary and sufficient conditions for a geometrically infinite surface to have a discrete spectrum. After deriving a number of properties of the length spectrum, we show that every topological surface of infinite type admits both an infinite dimensional family of quasiconformally distinct hyperbolic structures having a discrete length spectrum, and an infinite dimensional family of quasiconformally distinct structures with a nondiscrete spectrum. Moreover, there exists such an infinite dimensional subspace arbitrarily close to (in the Fenchel-Nielsen topology) any hyperbolic structure.