\Gamma-extensions of the spectrum of an orbifold

TL;DR

This paper introduces the mma-spectrum, an extension of the Laplace spectrum for orbifolds that captures singularities and helps distinguish isospectral orbifolds, with new construction methods for mma-isospectral pairs.

Contribution

It defines the mma-spectrum for orbifolds, compares it across examples, and proves a generalized Sunada's theorem for constructing mma-isospectral orbifold pairs.

Findings

mma-spectrum relates to orbifold singularities.

Isospectral orbifolds may not be mma-isospectral.

A generalized Sunada's theorem enables new mma-isospectral constructions.

Abstract

We introduce the \Gamma-extension of the spectrum of the Laplacian of a Riemannian orbifold, where \Gamma is a finitely generated discrete group. This extension, called the \Gamma-spectrum, is the union of the Laplace spectra of the \Gamma-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the \Gamma-spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that it many cases, isospectral orbifolds need not be \Gamma-isospectral. We additionally prove a version of Sunada's theorem that allows us to construct pairs of orbifolds that are \Gamma-isospectral for any choice of \Gamma.