$L^2$-Betti numbers of locally compact groups and their cross section equivalence relations
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- by David Kyed, Henrik Densing Petersen and Stefaan Vaes PDF
- Trans. Amer. Math. Soc. 367 (2015), 4917-4956 Request permission
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.References
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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
- MR Author ID: 854864
- Email: david.kyed@wis.kuleuven.be, dkyed@imada.sdu.dk
- 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
- MR Author ID: 1002082
- Email: hdp@math.ku.dk, henrik.petersen@epfl.ch
- Stefaan Vaes
- Affiliation: Department of Mathematics, KU Leuven, Leuven, Belgium
- Email: stefaan.vaes@wis.kuleuven.be
- 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. - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 4917-4956
- MSC (2010): Primary 22D40; Secondary 22F10, 28D15, 37A20
- DOI: https://doi.org/10.1090/S0002-9947-2015-06449-6
- MathSciNet review: 3335405