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Spectral Estimates for Towers of Noncompact Quotients

Published online by Cambridge University Press:  20 November 2018

Anton Deitmar  and
Werner Hoffman
Anton Deitmar
Affiliation:
Mathematisches Institut, Im Neuenheimer Feld 288, 69126 Heidelberg, Germany email: anton@mathi.uni-heidelberg.de
Werner Hoffman
Affiliation:
Humboldt-Universität zu Berlin, Institut für Mathematik, Jägerstr. 10/11, 10117 Berlin, Germany email: hoffmann@mathematik.hu-berlin.de
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Abstract

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We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Gamma$-automorphic Laplacian below a given bound when $\Gamma$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Gamma$ in the family is assumed to contain a principal congruence subgroup whose index in $\Gamma$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Gamma$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice $\Gamma$.

Keywords

11F7258G2522E40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 2 , 01 April 1999 , pp. 266 - 293
Copyright
Copyright © Canadian Mathematical Society 1999

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