# Isospectrality For Orbifold Lens Spaces

## TL;DR

This paper demonstrates that in 3- and 4-dimensional orbifold lens spaces, isospectrality does not imply geometric equivalence, and shows limitations of heat kernel coefficients in distinguishing such spaces.

## Contribution

It proves isospectrality does not determine geometry for orbifold lens spaces in all dimensions and highlights limitations of heat kernel asymptotics.

## Key findings

- Isospectral orbifold lens spaces can be non-isometric.
- Heat kernel asymptotic coefficients are insufficient to distinguish orbifold lens spaces.
- Complete answer to 'can one hear the shape of a drum?' for orbifold lens spaces.

## Abstract

We answer Mark Kac's famous question, "can one hear the shape of a drum?" in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all dimensions. We also show that the coefficients of the asymptotic expansion of the trace of the heat kernel are not sufficient to determine the above results.