Isospectral surfaces with distinct covering spectra via Cayley Graphs

TL;DR

This paper constructs examples of isospectral surfaces with different covering spectra using Cayley graphs, demonstrating that the covering spectrum is not determined by the spectrum alone.

Contribution

It provides the first known examples of isospectral surfaces with distinct covering spectra, using Cayley graphs and Sunada's method.

Findings

Existence of isospectral Cayley graphs with different covering spectra

Construction of infinitely many Sunada-isospectral surfaces with unequal covering spectra

Demonstration that the covering spectrum is not a spectral invariant

Abstract

The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we demonstrated that the covering spectrum is not a spectral invariant of a manifold in dimensions three and higher. In this article we give an example of two isospectral Cayley graphs that admit length space structures with distinct covering spectra. From this we deduce the existence of infinitely many pairs of Sunada-isospectral surfaces with unequal covering spectra.