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We prove that the L^2-Betti numbers of a unimodular locally compact group Gcoincide, up to a natural scaling constant, with the L^2-Betti numbers of thecountable equivalence relation induced on a cross section of any essentiallyfree ergodic probability measure preserving action of G. As a consequence, weobtain that the reduced and un-reduced L^2-Betti numbers of G agree and thatthe L^2-Betti numbers of a lattice Gamma in G equal those of G up to scaling bythe covolume of Gamma in G. We also deduce several vanishing results, includingthe vanishing of the reduced L^2-cohomology for amenable locally compactgroups.

L^2-Betti numbers of locally compact groups and their cross section equivalence relations