Orbifold Lens Spaces that are Isospectral but not Isometric

TL;DR

This paper demonstrates that in higher dimensions, orbifold lens spaces can be constructed to be isospectral yet not isometric, extending classical results to the orbifold setting and answering Kac's question negatively.

Contribution

The authors extend Ikeda's techniques to orbifold lens spaces, proving the existence of infinitely many isospectral non-isometric pairs in higher dimensions.

Findings

Existence of infinitely many isospectral non-isometric orbifold lens spaces in certain dimensions.

Generalization of results to all dimensions greater than 8.

Extension of classical spectral geometry results to orbifold settings.

Abstract

We answer Mark Kacs famous question - can one hear the shape of a drum - in the negative for orbifolds that are spherical space forms. This is done by extending the techniques developed by A. Ikeda on Lens Spaces to the orbifold setting. Several results are proved to show that with certain restrictions on the dimensionalities of orbifold Lens spaces we can obtain infinitely many pairs of isospectral non-isometric Lens spaces. These results are then generalized to show that for any dimension greater than 8 we can have pairs of isospectral non-isometric orbifold Lens spaces.