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

Friedrich Martin Schneider

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.

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

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.
Paper Structure (12 sections, 87 theorems, 322 equations)

This paper contains 12 sections, 87 theorems, 322 equations.

Table of Contents

  1. Introduction
  2. Concentration in uniform spaces
  3. Amenability and convolution algebras
  4. Concentration of invariant means
  5. Solution to a problem by Pestov
  6. Continuous geometries and their maximal chains
  7. Continuous rings and their maximal nests
  8. Algebraic elements in continuous rings
  9. Continuous triangularization
  10. Nest envelopes
  11. Dynamics of stable subgroups
  12. Nilpotency and Levitzki radical

Key Result

Theorem 1.1

Consider a chain of amenable topological groupseach equipped with the relative topology inherited from $G$ For each $i \in \{ 1,\ldots,n\}$ pick a $G_{i}$-left-invariant mean $\nu_{i} \in \mathop{\mathrm{M}}\nolimits (G_{i})$, and consider Moreover, let $d$ be a bounded, continuous, right-invariant pseudo-metric on $G$. Then, for every $f \in \mathop{\mathrm{Lip}}\nolimits_{1}(G,d)$ and every $\v

Theorems & Definitions (209)

  • Theorem 1.1: Theorem \ref{['theorem:main.concentration']}
  • Corollary 1.2: Proposition \ref{['proposition:amenable.folding']} $+$ Corollary \ref{['corollary:extreme.amenability.2']}
  • Theorem 1.4: Theorem \ref{['theorem:stable.groups']}
  • Corollary 1.5: Corollary \ref{['corollary:final']}
  • Corollary 1.6: Corollary \ref{['corollary:inert.final']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 199 more