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Transactions of the American Mathematical Society

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

   
 
 

 

Multiplication of weak equivalence classes may be discontinuous


Author: Anton Bernshteyn
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37A15, 22F10, 37A35; Secondary 20G40
DOI: https://doi.org/10.1090/tran/7847
Published electronically: June 13, 2019
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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}}_{\... ... } \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 \geq 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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Anton Bernshteyn
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Illinois; and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania
Email: bernsht2@illinois.edu; abernsht@math.cmu.edu

DOI: https://doi.org/10.1090/tran/7847
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.
Article copyright: © Copyright 2019 Anton Bernshteyn