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A SPECTRAL GAP PROPERTY FOR SUBGROUPS OF FINITE COVOLUME

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 λG/H of G on L(G/H) has a spectral gap, that is, the restriction of λG/H to the orthogonal complement of the constants in L(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.