A Uniform Strong Spectral Gap for Congruence Covers of a compact quotient of PSL(2,R)^d

2010/02/18 by Dubi Kelmer, Kelmer, Dubi · 1 citation
Chemistry · Mathematics · #Advanced Algebra and Geometry #Crystal structures of chemical compounds #Finite Group Theory Research #math.NT #math.SP #msc:11F72 #msc:11M36 #msc:22E45

paper · pdf · doi:10.48550/arxiv.1002.3422

22 pages, Order and numbering of theorems has changed. Final version, to appear in IMRN

arxiv created 2010/05/19 · arxiv updated 2010/05/21

Abstract

The existence of a strong spectral gap for lattices in semi-simple Lie groups is crucial in many applications. In particular, for arithmetic lattices it is useful to have bounds for the strong spectral gap that are uniform in the family of congruence covers. When the lattice is itself a congruence group, there are uniform and very good bounds for the spectral gap coming from the known bounds toward the Ramanujan-Selberg conjectures. In this note, we establish a uniform bound for the strong spectral gap for congruence covers of an irreducible co-compact lattice Γ in PSL(2,R)^d with d≥ 2, which is the simplest and most basic case where the congruence subgroup property is not known.

Cited by

• The density Hypothesis for irreducible lattices in PSL₂(ℝ)^d

Related

• Strong Spectral Gaps for Compact Quotients of Products of \PSL(2,\bbR)
• A uniform spectral gap for congruence covers of a hyperbolic manifold
• Quantitative spectral gap for thin groups of hyperbolic isometries
• The density Hypothesis for irreducible lattices in PSL₂(ℝ)^d
• Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces