Open Access
Fall 2009 Cheeger constants of arithmetic hyperbolic 3-manifolds
Dominic Lanphier, Jason Rosenhouse
Illinois J. Math. 53(3): 769-783 (Fall 2009). DOI: 10.1215/ijm/1286212915

Abstract

We study the Cheeger constants of certain infinite families of arithmetic hyperbolic three-manifolds, as well as certain graphs associated to these manifolds. We derive computable bounds on the Cheeger constants, and therefore bounds on the first eigenvalue of the Laplacian, by adapting discrete methods due to Brooks, Perry and Petersen. We then modify probabilistic methods due to Brooks and Zuk to obtain sharper, asymptotic bounds. A consequence is that the Cheeger constants are quite small, implying that Cheeger’s inequality is generally insufficient to prove Selberg’s eigenvalue conjecture.

Citation

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Dominic Lanphier. Jason Rosenhouse. "Cheeger constants of arithmetic hyperbolic 3-manifolds." Illinois J. Math. 53 (3) 769 - 783, Fall 2009. https://doi.org/10.1215/ijm/1286212915

Information

Published: Fall 2009
First available in Project Euclid: 4 October 2010

zbMATH: 1207.58024
MathSciNet: MR2727354
Digital Object Identifier: 10.1215/ijm/1286212915

Subjects:
Primary: 58J50

Rights: Copyright © 2009 University of Illinois at Urbana-Champaign

Vol.53 • No. 3 • Fall 2009
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