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

TL;DR

This paper proves a uniform strong spectral gap for congruence covers of a compact quotient of PSL(2,R)^d, advancing understanding in spectral theory for non-congruence lattices in semi-simple Lie groups.

Contribution

It establishes the first uniform spectral gap bound for congruence covers of irreducible co-compact lattices in PSL(2,R)^d where the congruence subgroup property is unknown.

Findings

Proves a uniform spectral gap for congruence covers in PSL(2,R)^d

Advances spectral gap bounds in non-congruence lattice cases

Provides foundational results for applications in automorphic forms and number theory

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 $\Gamma$ in $PSL(2,R)^d$ with $d\geq 2$, which is the simplest and most basic case where the congruence subgroup property is not known.