Isospectral hyperbolic surfaces of infinite genus

TL;DR

This paper demonstrates the existence of large families of hyperbolic surfaces with infinite genus and specific end structures that are isospectral, meaning they share the same spectral properties despite being geometrically distinct.

Contribution

It constructs uncountably many isospectral hyperbolic structures on infinite-type surfaces with self-similar ends, advancing understanding of spectral geometry in infinite genus contexts.

Findings

Existence of arbitrarily large isospectral families for surfaces without planar ends.

Construction of uncountably many isospectral, quasiconformally distinct structures with self-similar ends.

Extension of spectral geometry results to infinite genus hyperbolic surfaces.

Abstract

We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable family of isospectral and quasiconformally distinct hyperbolic structures.