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

Abstract

We prove that the L 2 L^2 -Betti numbers of a unimodular locally compact group G G coincide, up to a natural scaling constant, with the L 2 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 G . As a consequence, we obtain that the reduced and unreduced L 2 L^2 -Betti numbers of G G agree and that the L 2 L^2 -Betti numbers of a lattice Γ \Gamma in G G equal those of G G up to scaling by the covolume of Γ \Gamma in G G . We also deduce several vanishing results, including the vanishing of the reduced L 2 L^2 -cohomology for amenable locally compact groups.