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

Abstract

Abstract. The existence of a strong spectral gap for quotients \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 RamanujanSelberg 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 in G = PSL(2, R) d for d ≥ 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.

Distribution of holonomy about closed geodesics in a product of hyperbolic planes

“…∈ [0, 1 2 ) to measure the spectral gap for V m and let α = sup m α m . The results of Kelmer and Sarnak [7] imply that α m < 0.34 for all but finitely many values of m; in particular we have that α < 1 2 .…”

A Uniform Strong Spectral Gap for Congruence Covers of a Compact Quotient of

“…Such an upper bound can indeed be obtained by refining the analysis in [KeSa]. Specifically, following the arguments in the proof of [KeSa,Theorem 4] adapted to congruence covers, we give an alternative upper bound for the multiplicities.…”

Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces

“…For higher rank cases, Deitmar [1] defined and studied "generalized Selberg zeta functions" for compact higher rank locally symmetric spaces. (See also Kelmer-Sarnak [15]). Therefore, our concern is to define and study "Selberg type zeta functions" for noncompact higher rank locally symmetric spaces such as Hilbert modular surfaces.…”