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 Γ in G equal those of G up to scaling by the covolume of Γ in G. We also deduce several vanishing results, including the vanishing of the reduced L 2 -cohomology for amenable locally compact groups.

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The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute to the general theory of L2‐Betti numbers of totally disconnected groups and provide exact computations of the L2‐Betti numbers of the Neretin group and Chevalley groups over the field of Laurent series over a finite field and their lattices.

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