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Proceedings of the American Mathematical Society

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Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues


Authors: Brian A. Benson, Grant S. Lakeland and Holger Then
Journal: Proc. Amer. Math. Soc. 149 (2021), 417-438
MSC (2010): Primary 57M50; Secondary 58C40
DOI: https://doi.org/10.1090/proc/15152
Published electronically: October 9, 2020
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Abstract: We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-cocompact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp forms. From geometrical considerations, we find evidence for the existence of small eigenvalues. We search for these small eigenvalues and compute the corresponding Maass cusp forms numerically.


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Additional Information

Brian A. Benson
Affiliation: Department of Mathematics, University of California, Riverside, 900 University Avenue, Riverside, California 92521
Email: bbenson@ucr.edu

Grant S. Lakeland
Affiliation: Department of Mathematics & Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston, Illinois 61920
Email: gslakeland@eiu.edu

Holger Then
Affiliation: Freie Waldorfschule Augsburg, Dr.-Schmelzing-Straße 52, 86169 Augsburg, Germany
Email: holger.then@gmx.de

DOI: https://doi.org/10.1090/proc/15152
Received by editor(s): August 9, 2019
Received by editor(s) in revised form: April 23, 2020, and April 25, 2020
Published electronically: October 9, 2020
Communicated by: David Futer
Article copyright: © Copyright 2020 American Mathematical Society