Isospectrality for infinite-type hyperbolic surfaces with discrete length spectrum

TL;DR

This paper proves finiteness of isospectral hyperbolic surfaces with discrete length spectrum from Sunada's method and constructs examples with infinite symmetry groups and unbounded families under certain topological conditions.

Contribution

It establishes the finiteness of isospectral families from Sunada's method and introduces topological concepts to realize any finite group as an isometry group of such surfaces.

Findings

Isospectral families from Sunada's method are finite.

Any finite group can be realized as the isometry group of a hyperbolic surface with discrete spectrum.

Isospectral families can have unbounded size within a fixed moduli space.

Abstract

We prove that every family of isospectral surfaces with discrete length spectrum arising from Sunada's method is finite. Furthermore, by introducing the topological notion of surfaces with self-duplicating ends, we show that every finite group can be realized as the full isometry group of a hyperbolic structure with discrete spectrum on such a surface, if the genus is infinite. Under the same topological assumptions, we also demonstrate that the above-mentioned isospectral families can have unbounded cardinality within a fixed moduli space.