L²-Betti numbers of locally compact groups and their cross section equivalence relations

We prove that the L^2-Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L^2-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and un-reduced L^2-Betti numbers of G agree and that the L^2-Betti numbers of a lattice Gamma in G equal those of G up to scaling by the covolume of Gamma in G. We also deduce several vanishing results, including the vanishing of the reduced L^2-cohomology for amenable locally compact groups.