Spectral Estimates for Towers of Noncompact Quotients

http://dx.doi.org/10.4153/CJM-1999-014-3
Canad. J. Math. 51(1999), 266-293

  Published:1999-04-01
 Printed: Apr 1999

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~$\Ga$.