Strong spectral gaps for compact quotients of products of PSL(2,ℝ)

  • Peter Sarnak

    Princeton University, United States
  • Dubi Kelmer

    University of Chicago, United States


The existence of a strong spectral gap for quotients Γ\bsG\Gamma\bs G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If GG has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice Γ\Gamma in G=\PSL(2,\bbR)dG=\PSL(2,\bbR)^d for d2d\geq 2 which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.

Cite this article

Peter Sarnak, Dubi Kelmer, Strong spectral gaps for compact quotients of products of PSL(2,ℝ). J. Eur. Math. Soc. 11 (2009), no. 2, pp. 283–313

DOI 10.4171/JEMS/151