$\Gamma$-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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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