The fundamental group of a spherical space form is not audible

TL;DR

This paper demonstrates that the fundamental group of a spherical space form cannot always be determined from its Laplace spectrum, providing counterexamples and new insights into spectral geometry.

Contribution

It presents the first known pair of spherical space forms with non-isomorphic fundamental groups sharing the same spectrum, challenging previous assumptions.

Findings

Identified the first pair of isospectral spherical space forms with different fundamental groups.

Showed that the fundamental group is not always audible in spherical space forms.

Found instances where the fundamental group can be determined from the spectrum.

Abstract

We revisit the problem of isospectral spherical space forms with non-cyclic fundamental groups after the works by Ikeda, Gilkey and Wolf. We find the first pair of spherical space forms with non-isomorphic fundamental groups and the same Laplace spectrum. This shows that the isomorphism class of the fundamental group is not audible among spherical space forms. We also found several instances where one can hear the fundamental group of a spherical space form (among spherical space forms).