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

Authors: David Kyed, Henrik Densing Petersen and Stefaan Vaes
Journal: Trans. Amer. Math. Soc. 367 (2015), 4917-4956
MSC (2010): Primary 22D40; Secondary 22F10, 28D15, 37A20
Published electronically: January 29, 2015
MathSciNet review: 3335405
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Abstract: 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 $ \Gamma $ in $ G$ equal those of $ G$ up to scaling by the covolume of $ \Gamma $ 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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Additional Information

David Kyed
Affiliation: Department of Mathematics, KU Leuven, Leuven, Belgium
Address at time of publication: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

Henrik Densing Petersen
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
Address at time of publication: SB-MATHGEOM-EGG, EPFL, Station 8, CH-1015, Lausanne, Switzerland

Stefaan Vaes
Affiliation: Department of Mathematics, KU Leuven, Leuven, Belgium

Received by editor(s): April 20, 2013
Published electronically: January 29, 2015
Additional Notes: The first author was supported by ERC Starting Grant VNALG-200749
The second author was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92)
The third author was supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the Research Foundation – Flanders (FWO), and KU Leuven BOF research grant OT/08/032.
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