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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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Multiplication of weak equivalence classes may be discontinuous
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by Anton Bernshteyn PDF
Trans. Amer. Math. Soc. 372 (2019), 8091-8106

Abstract:

For a countably infinite group $\Gamma$, let ${\mathcal {W}}_{\Gamma }$ denote the space of all weak equivalence classes of measure-preserving actions of ${\Gamma }$ on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Abért and Elek. There is a natural multiplication operation on ${\mathcal {W}}_{\Gamma }$ (induced by taking products of actions) that makes ${\mathcal {W}}_{\Gamma }$ an Abelian semigroup. Burton, Kechris, and Tamuz showed that if ${\Gamma }$ is amenable, then ${\mathcal {W}}_{\Gamma }$ is a topological semigroup; i.e., the product map ${\mathcal {W}}_{\Gamma } \times {\mathcal {W}}_{\Gamma } \to {\mathcal {W}}_{\Gamma } \colon (\mathfrak {a}, \mathfrak {b}) \mapsto \mathfrak {a} \times \mathfrak {b}$ is continuous. In contrast to that, we prove that if ${\Gamma }$ is a Zariski dense subgroup of ${\mathrm {SL}}_d({\mathbb {Z}})$ for some $d \geqslant 2$ (for instance, if ${\Gamma }$ is a non-Abelian free group), then multiplication on ${\mathcal {W}}_{\Gamma }$ is discontinuous, even when restricted to the subspace ${\mathcal {FW}}_{\Gamma }$ of all free weak equivalence classes.
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Additional Information
  • Anton Bernshteyn
  • Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Illinois; and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania
  • MR Author ID: 1104079
  • Email: bernsht2@illinois.edu; abernsht@math.cmu.edu
  • Received by editor(s): April 4, 2018
  • Received by editor(s) in revised form: March 11, 2019
  • Published electronically: June 13, 2019
  • Additional Notes: This research is supported in part by the Waldemar J., Barbara G., and Juliette Alexandra Trjitzinsky Fellowship.
  • © Copyright 2019 Anton Bernshteyn
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 8091-8106
  • MSC (2010): Primary 37A15, 22F10, 37A35; Secondary 20G40
  • DOI: https://doi.org/10.1090/tran/7847
  • MathSciNet review: 4029691