  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.
