Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this tec...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 33; no. 6; pp. 1608 - 1681
Main Author Schneider, Friedrich Martin
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2023
Springer Nature B.V
Subjects
Analysis
Chain dynamics
Invariants
Martingales
Mathematics
Mathematics and Statistics
Subgroups
Topology
Concentration of measure
continuous ring
06C20
16E50
22A10
topological group
continuous geometry
extreme amenability
43A07
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ISSN1016-443X
1420-8970
DOI10.1007/s00039-023-00651-w

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Summary:We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-023-00651-w