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

David Kyed (University of Leuven)

Tuesday 7th May, 2013 16:00-17:00 Maths 203


In a recent joint work with Petersen and Vaes we prove that  the L2-Betti numbers of a locally compact group coincide, up to a natural scaling 
constant, with the L2-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic pmp action
of it. As a consequence, we obtain that the reduced and un-reduced L2-Betti numbers agree and that the L2-Betti numbers of any lattice equal
the ones for the ambient group scaled with the covolume of the lattice in question. Furthermore, several vanishing results are derived from this,
including the vanishing of the reduced L2-cohomology for amenable groups. In my talk I will elaborate on these results and survey the theory
of L2-Betti numbers and cross section equivalence relations.

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