A spectral gap property for subgroups of finite covolume in Lie groups
Volume 118 / 2010
Colloquium Mathematicum 118 (2010), 175-182
MSC: 22E40, 37A30, 43A85.
DOI: 10.4064/cm118-1-9
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 complement 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.
