Critical exponents of discrete groups and 𝐿²–spectrum

Leuzinger, Enrico

doi:10.1090/S0002-9939-03-07173-9

Critical exponents of discrete groups and $L^2$–spectrum
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by Enrico Leuzinger

Proc. Amer. Math. Soc. 132 (2004), 919-927

DOI: https://doi.org/10.1090/S0002-9939-03-07173-9

Published electronically: September 12, 2003

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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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