Γ-extensions of the spectrum of an orbifold

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

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