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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Critical exponents of discrete groups and $L^2$-spectrum


Author: Enrico Leuzinger
Journal: Proc. Amer. Math. Soc. 132 (2004), 919-927
MSC (2000): Primary 22E40, 53C20, 53C35
Published electronically: September 12, 2003
MathSciNet review: 2019974
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Abstract: Let $G$ be a noncompact semisimple Lie group and $\Gamma$ an arbitrary discrete, torsion-free subgroup of $G$. Let $\lambda_0(M)$ be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space $M=\Gamma\backslash X$, and let $\delta(\Gamma)$ be the exponent of growth of $\Gamma$. If $G$ has rank $1$, then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating $\lambda_0(M)$ from above and below by quadratic polynomials in $\delta(\Gamma)$. As an application we prove a rigiditiy property of lattices.


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Additional Information

Enrico Leuzinger
Affiliation: Math. Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Email: Enrico.Leuzinger@math.uni-karlsruhe.de

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07173-9
PII: S 0002-9939(03)07173-9
Keywords: Discrete subgroups of semisimple Lie groups, critical exponent, $L^2$--spectrum, locally symmetric spaces
Received by editor(s): November 9, 2002
Published electronically: September 12, 2003
Communicated by: Rebecca A. Herb
Article copyright: © Copyright 2003 American Mathematical Society