## $\Gamma$-extensions of the spectrum of an orbifold

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- by Carla Farsi, Emily Proctor and Christopher Seaton PDF
- Trans. Amer. Math. Soc.
**366**(2014), 3881-3905 Request permission

## 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, in 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 $\Gamma$.

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## Additional Information

**Carla Farsi**- Affiliation: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395
- MR Author ID: 311031
- Email: farsi@euclid.colorado.edu
**Emily Proctor**- Affiliation: Department of Mathematics, Middlebury College, Middlebury, Vermont 05753
- Email: eproctor@middlebury.edu
**Christopher Seaton**- Affiliation: Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, Tennessee 38112
- MR Author ID: 788748
- Email: seatonc@rhodes.edu
- Received by editor(s): August 20, 2012
- Received by editor(s) in revised form: December 12, 2012
- Published electronically: December 27, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**366**(2014), 3881-3905 - MSC (2010): Primary 58J53, 57R18; Secondary 53C20
- DOI: https://doi.org/10.1090/S0002-9947-2013-06082-5
- MathSciNet review: 3192622