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

TL;DR

This paper establishes a concentration inequality for invariant means on topological groups, leveraging martingale techniques to identify new extremely amenable groups from continuous geometries and addressing a question on dynamical concentration.

Contribution

It introduces a novel concentration inequality for invariant means, applies it to continuous geometries, and provides new examples of extremely amenable groups, advancing understanding of group dynamics.

Findings

Proved a concentration inequality for invariant means.

Identified new extremely amenable groups from continuous geometries.

Answered a question on dynamical concentration in product groups.

Abstract

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.