A Bound on Length Isospectral Families of Hyperbolic Surfaces

TL;DR

This paper establishes a bound on the number of hyperbolic surfaces that are length isospectral to a given surface, depending only on topological features and the shortest geodesic length, advancing understanding of spectral geometry.

Contribution

It provides a new bound on isospectral hyperbolic surfaces based on topological type and systole length, extending previous results to broader classes of surfaces.

Findings

Bound depends only on topological type and shortest geodesic length

Applicable to families with Bers' constant and systole length bounds

Advances spectral geometry understanding

Abstract

In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed geodesic on the surface. This will follow from a more general bound applying to any family of hyperbolic surfaces which admits a Bers' constant and with a lower bound on systole length.