A SPECTRAL GAP PROPERTY FOR SUBGROUPS OF FINITE COVOLUME IN LIE GROUPS

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.