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15 September 2020 Invariant means and the structure of inner amenable groups
Robin D. Tucker-Drob
Duke Math. J. 169(13): 2571-2628 (15 September 2020). DOI: 10.1215/00127094-2019-0070

Abstract

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G . Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first 2 -Betti number of G with that of the stabilizer subgroups. In addition, for any marked finitely generated nonamenable group G we establish a uniform isoperimetric threshold for Schreier graphs G / H of G , beyond which the group H is necessarily weakly normal in G .

Even more can be said in the particular case of an atomless mean for the conjugation action—that is, when G is inner amenable. We show that inner amenable groups have fixed price 1 , and we establish cocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field.

As a special case of inner amenability, we consider groups which are stable in the sense of Jones and Schmidt, obtaining a complete characterization of linear groups which are stable. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H ( R ) , of piecewise-projective homeomorphisms of the line, are stable.

Citation

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Robin D. Tucker-Drob. "Invariant means and the structure of inner amenable groups." Duke Math. J. 169 (13) 2571 - 2628, 15 September 2020. https://doi.org/10.1215/00127094-2019-0070

Information

Received: 24 April 2018; Revised: 22 July 2019; Published: 15 September 2020
First available in Project Euclid: 1 September 2020

MathSciNet: MR4142752
Digital Object Identifier: 10.1215/00127094-2019-0070

Subjects:
Primary: 37A20
Secondary: 20H20, ‎43A07‎

Rights: Copyright © 2020 Duke University Press

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Vol.169 • No. 13 • 15 September 2020
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