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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Carla Farsi, Emily Proctor and Christopher Seaton
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
Published electronically: December 27, 2013
MathSciNet review: 3192622
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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

Keywords: Orbifold, isospectral, twisted sector, spectral geometry
Received by editor(s): August 20, 2012
Received by editor(s) in revised form: December 12, 2012
Published electronically: December 27, 2013
Article copyright: © Copyright 2013 American Mathematical Society