Strong Spectral Gaps for Compact Quotients of Products of $\PSL(2,\bbR)$

TL;DR

This paper establishes effective bounds for the spectral gap of irreducible co-compact lattices in products of , advancing understanding in cases lacking known congruence subgroup properties.

Contribution

It provides the first effective bounds for spectral gaps in  quotients where the congruence subgroup property is unknown.

Findings

Effective bounds for spectral gaps in  cases

Advances understanding of spectral gaps without congruence subgroup property

Applicable to irreducible co-compact lattices in 

Abstract

The existence of a strong spectral gap for quotients $\Gamma\bs G$ 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 $G$ has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible co-compact lattice $\Gamma$ in $G=\PSL(2,\bbR)^d$ 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.