Isospectral Finiteness of Hyperbolic Orbisurfaces

Abstract

We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the obstructions involve the number of singular points and genera of our orbisurfaces. Using a version of the Selberg Trace Formula for hyperbolic orbisurfaces, we show that the Laplace spectrum determines the length spectrum and the orders of the singular points, up to finitely many possibilities. Conversely, knowledge of the length spectrum and the orders of the singular points determines the Laplace spectrum. This partial generalization of Huber's theorem is used to prove that isospectral sets of hyperbolic orbisurfaces have finite cardinality, generalizing a result of McKean for Riemann surfaces.