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

Dubi Kelmer, Peter Sarnak

Type: Article

Publication Date: 2009-04-30

Citations: 28

DOI: https://doi.org/10.4171/jems/151

Abstract

The existence of a strong spectral gap for quotients \Gamma\backslash 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, 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 \Gamma in G=\operatorname{PSL}(2,\mathbb R)^d for d\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.

Locations

Journal of the European Mathematical Society - View - PDF
arXiv (Cornell University) - PDF
Journal of the European Mathematical Society - View - PDF
arXiv (Cornell University) - PDF

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Works Cited by This (46)

Action	Title	Year	Authors
+	Notes on the Generalized Ramanujan Conjectures	2005	Peter Sarnak
+	The bottom of the spectrum of a Riemannian covering.	1985	Robert Brooks
+	Sur les coefficients des representations unitaires des groupes de Lie simples	1979	Michael Cowling
+	Algebras with Involutions and the Classical Groups	1960	André Weil
+	Zur Zahlentheorie der Quaternionen-Algebren.	1955	Martin Eichler
+	Algebraic Number Theory	1999	Jürgen Neukirch
+ PDF	Book Review: The Selberg trace formula for $PSL\,\left( {2,\mathbf{R}} \right)$, Volume I	1978	S. J. Patterson
+	Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants	2002	Hee Oh
+	Related aspects of positivity in Riemannian geometry	1987	Dennis Sullivan
+ PDF	Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups	2000	A. Borel Nolan R. Wallach