Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 22E40, 53C20, 53C35

Retrieve articles in all journals with MSC (2000): 22E40, 53C20, 53C35

Additional Information

Enrico Leuzinger
Affiliation: Math. Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany

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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia