A spectral gap property for subgroups of finite covolume in Lie groups

TL;DR

This paper proves that for certain subgroups of Lie groups with finite covolume, the associated unitary representation exhibits a spectral gap, impacting the understanding of spectral properties of infinite volume locally symmetric spaces.

Contribution

It establishes a spectral gap property for unitary representations of Lie groups on quotient spaces with finite covolume, answering a question by G. Margulis.

Findings

Unitary representation lambda_{G/H} has a spectral gap.

No almost invariant vectors in the orthogonal complement of constants.

Applications to spectral geometry of infinite volume spaces.

Abstract

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation lambda_{G/H} of G on L^2(G/H) has a spectral gap, that is, the restriction of lambda_{G/H} to the orthogonal of the constants in L^2(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.