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

Dubi Kelmer, Peter Sarnak

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17 Scopus citations


The existence of a strong spectral gap for quotients L\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 G has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice L in G D PSL(2,ℝ)d for d ge; 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.

Original languageEnglish (US)
Pages (from-to)283-313
Number of pages31
JournalJournal of the European Mathematical Society
Issue number2
StatePublished - 2009

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


  • Product of hyperbolic planes
  • Selberg trace formula
  • Spectral gap


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