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On inner-amenability and boundary actions

Jacopo Bassi

TL;DR

The paper investigates inner-amenability and boundary actions for discrete groups, focusing on the interplay between inner-amenability, the AO-property, and bi-exactness. It shows that for ICC and non-amenable groups, inner-amenability precludes the AO-property by leveraging a non-standard boundary extension and temperedness criteria for the Calkin representation. Additionally, it proves that discrete ICC subgroups of products of lcsc bi-exact groups cannot be inner-amenable under natural projection or irreducibility assumptions, extending known non-bi-exactness results. The methods connect boundary dynamics, tempered representations, and spectral-gap considerations to yield rigidity results with implications for the structure of group C*-algebras and von Neumann algebras.

Abstract

Let $Γ$ be a discrete countable group. The first main result of this work is that if $Γ$ is ICC inner-amenable non-amenable then it cannot satisfy the (AO)-property, answering a question posed by C. Anantharaman-Delaroche. It is also proved that if $Γ$ is a "sufficiently large" discrete subgroup of a product of locally compact second countable bi-exact groups, then it cannot be inner-amenable. Both these results generalize the well-known fact that ICC non-amenable inner-amenable discrete countable groups cannot be bi-exact.

On inner-amenability and boundary actions

TL;DR

The paper investigates inner-amenability and boundary actions for discrete groups, focusing on the interplay between inner-amenability, the AO-property, and bi-exactness. It shows that for ICC and non-amenable groups, inner-amenability precludes the AO-property by leveraging a non-standard boundary extension and temperedness criteria for the Calkin representation. Additionally, it proves that discrete ICC subgroups of products of lcsc bi-exact groups cannot be inner-amenable under natural projection or irreducibility assumptions, extending known non-bi-exactness results. The methods connect boundary dynamics, tempered representations, and spectral-gap considerations to yield rigidity results with implications for the structure of group C*-algebras and von Neumann algebras.

Abstract

Let be a discrete countable group. The first main result of this work is that if is ICC inner-amenable non-amenable then it cannot satisfy the (AO)-property, answering a question posed by C. Anantharaman-Delaroche. It is also proved that if is a "sufficiently large" discrete subgroup of a product of locally compact second countable bi-exact groups, then it cannot be inner-amenable. Both these results generalize the well-known fact that ICC non-amenable inner-amenable discrete countable groups cannot be bi-exact.
Paper Structure (4 sections, 4 theorems, 1 equation)

This paper contains 4 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Inner-amenability and (AO)-property
  3. Discrete subgroups of products of bi-exact groups
  4. Acknowledgments

Key Result

Theorem 2.1

Let $\Gamma$ be a discrete countable group acting on a countable set $X$ and suppose that the Calkin representation $C^*\Gamma \rightarrow \mathbb{B}(l^2 X)/\mathbb{K}(l^2 X)$ is tempered. Let $\Lambda$ be a subgroup of $\Gamma$ which fixes a probability measure on $\partial_\beta X$. Then $\Lambda$

Theorems & Definitions (5)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 3.1
  • Example 3.1