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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues
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by Brian A. Benson, Grant S. Lakeland and Holger Then PDF
Proc. Amer. Math. Soc. 149 (2021), 417-438 Request permission


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
  • MR Author ID: 892840
  • Email:
  • Grant S. Lakeland
  • Affiliation: Department of Mathematics & Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston, Illinois 61920
  • MR Author ID: 984963
  • Email:
  • Holger Then
  • Affiliation: Freie Waldorfschule Augsburg, Dr.-Schmelzing-Straße 52, 86169 Augsburg, Germany
  • MR Author ID: 742378
  • ORCID: 0000-0002-0368-639X
  • Email:
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 417-438
  • MSC (2010): Primary 57M50; Secondary 58C40
  • DOI:
  • MathSciNet review: 4172617