On G-sets and Isospectrality

TL;DR

This paper investigates the relationship between finite G-sets, their tensor products with manifolds, and the spectral properties of quotient spaces, revealing conditions under which non-isometric covers are isospectral.

Contribution

It demonstrates that manifolds with certain fundamental group quotients admit isospectral non-isometric covers, advancing understanding of spectral geometry and covering space relationships.

Findings

Manifolds with non-cyclic finite quotients have isospectral non-isometric covers.

Results connect group quotients to spectral properties of manifold covers.

Provides new examples of isospectral but non-isometric spaces.

Abstract

We study finite G-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: if M is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then M has isospectral non-isometric covers.