Isospectrality and isometry groups for infinite-type hyperbolic surfaces with discrete length spectrum
arXiv:2602.19670
The paper studies infinite‑type hyperbolic surfaces with a discrete length spectrum, showing that Sunada’s method yields only finite isospectral families but that arbitrarily large families exist, and proving that any finite group can be realized as the isometry group of such a surface.
Abstract
We study infinite-type hyperbolic surfaces with discrete length spectrum. In this setup, we show that Sunada's method can only produce finite isospectral families, but that there is no bound on the cardinality of an isospectral family (for any infinite-type surface without planar ends). We then prove that, given any infinite-genus surface satisfying an additional topological assumption, any finite group can be realized as isometry group of a hyperbolic structure with discrete length spectrum.
Generalized the construction of large isospectral families to all topological types. 13 pages, 4 figures