Isospectral spherical space forms and orbifolds of highest volume

TL;DR

This paper establishes the maximum possible volume for pairs of isospectral, non-isometric spherical orbifolds and space forms in various dimensions, revealing specific volume bounds based on dimension and congruence conditions.

Contribution

It provides explicit upper bounds on the volume of isospectral, non-isometric spherical orbifolds and space forms across different dimensions, extending the understanding of spectral geometry in these contexts.

Findings

Maximum volume of orbifolds in dimension d is vol(S^d)/8.

Maximum volume of space forms in certain dimensions is vol(S^{2n-1})/11.

Specific volume bounds depend on dimension and congruence conditions.

Abstract

We prove that $\operatorname{vol}(S^{d})/8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $d\geq5$. Furthermore, we show that $\operatorname{vol}(S^{2n-1})/11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $n\geq11$ and $n\equiv 1\pmod 5$, or $n\geq7$ and $n\equiv 2\pmod 5$, or $n\geq3$ and $n\equiv 3\pmod 5$.