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Spectral Estimates for Towers of Noncompact Quotients

 Printed: Apr 1999
  • Anton Deitmar
  • Werner Hoffman
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We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ 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~$\Ga$.
MSC Classifications: 11F72, 58G25, 22E40 show english descriptions Spectral theory; Selberg trace formula
unknown classification 58G25
Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
11F72 - Spectral theory; Selberg trace formula
58G25 - unknown classification 58G25
22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

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