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Stationary boundaries on the space of amenable subgroups and C*-simplicity

Anna Cascioli, Martín Gilabert Vio, Eduardo Silva

Abstract

We give a sufficient condition for a countable group $G$ to possess a probability measure $μ$ that admits a non-trivial $μ$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $μ$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $μ$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $μ\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $μ$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $μ$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries.

Stationary boundaries on the space of amenable subgroups and C*-simplicity

Abstract

We give a sufficient condition for a countable group to possess a probability measure that admits a non-trivial -boundary modeled in the space of amenable subgroups of . In particular, for such the space is not uniquely -stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group is C*-simple if and only if there exists such that is uniquely -stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group , whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial -boundary modeled on is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries.

Paper Structure

This paper contains 13 sections, 17 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Background and discussion
  4. SRSs that are not $\mu$-boundaries
  5. Existence of amenable boundary $\mu$-stationary random subgroups
  6. A short proof for wreath products
  7. Record times and the proof of Theorem \ref{['thm: syndetic']}
  8. Amenable boundary $\mu$-SRSs for Thompson's group $F$
  9. Amenable boundary $\mu$-SRSs for permutational wreath products
  10. Boundary SRSs are supported on normalish subgroups
  11. The proof of Theorem \ref{['thm: normalish']}
  12. Baumslag-Solitar groups and normalish subgroups
  13. Property (CS)

Key Result

Theorem 1

Let $G$ be a countable group. Suppose that there exists a non-trivial subgroup $H$ such that for all finite subsets $Q, Z \subseteq G$ there is $b \in G$ such that both $bZ$ and $b^{-1}Z$ are contained in Then there exists a non-degenerate, symmetric and finite-entropy probability measure $\mu$ on $G$ such that $\overline{\mathrm{Orb}_G(H)}$ supports a $\mu$-boundary SRS distinct from $\delta_{\{

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.1
  • ...and 33 more