Isospectral orbifolds with different maximal isotropy orders

TL;DR

This paper constructs pairs of compact Riemannian orbifolds that are isospectral yet differ in the maximal isotropy order of their singular points, revealing new phenomena in spectral geometry.

Contribution

It introduces novel examples of isospectral orbifolds with differing maximal isotropy orders, expanding understanding of spectral properties and isotropy in orbifolds.

Findings

Constructed orbifold pairs with different maximal isotropy orders that are isospectral.

Demonstrated isospectrality on functions using Sunada-type methods and dimension formulas.

Provided additional examples of isospectral orbifolds with varied properties.

Abstract

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on $p$-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean $\R^3$ and are shown to be isospectral on functions using dimension formulas for the eigenspaces. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties.