TY - JOUR
T1 - L2-Betti numbers of locally compact groups and their cross section equivalence relations
AU - Kyed, David
AU - Petersen, Henrik Densing
AU - Vaes, Stefaan
PY - 2015
Y1 - 2015
N2 - 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 unreduced 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.
AB - 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 unreduced 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.
UR - https://arxiv.org/abs/1302.6753
U2 - 10.1090/S0002-9947-2015-06449-6
DO - 10.1090/S0002-9947-2015-06449-6
M3 - Journal article
SN - 0002-9947
VL - 367
SP - 4917
EP - 4956
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 7
ER -