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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$L^2$-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.
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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