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Non-strong ergodicity of canonical actions of the Thompson groups

Ryoya Arimoto

TL;DR

This work addresses the strong ergodicity of canonical actions by Thompson-type groups on Cantor-like spaces and the consequent fullness of associated crossed products. It proves a general theorem: for any topologically principal, amenable, ample groupoid $\mathcal{G}$, the canonical action of its topological full group $[[\mathcal{G}]]$ on the unit space $\mathcal{G}^{(0)}$ is not strongly ergodic with respect to any quasi-invariant measure $\mu$. As a result, the crossed product von Neumann algebras $L^{\infty}(\mathcal{G}^{(0)},\mu) \bar{\rtimes} [[\mathcal{G}]]$ fail to be full, yielding non-embedding constraints for certain subgroups (e.g., copies of free groups) into these full groups. The theorem specializes to the Higman–Thompson and Brin–Thompson families and links amenability of the orbit equivalence relation to non-strong ergodicity via the germ groupoid framework, providing a broad mechanism to study non-full crossed products in this setting.

Abstract

We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.

Non-strong ergodicity of canonical actions of the Thompson groups

TL;DR

This work addresses the strong ergodicity of canonical actions by Thompson-type groups on Cantor-like spaces and the consequent fullness of associated crossed products. It proves a general theorem: for any topologically principal, amenable, ample groupoid , the canonical action of its topological full group on the unit space is not strongly ergodic with respect to any quasi-invariant measure . As a result, the crossed product von Neumann algebras fail to be full, yielding non-embedding constraints for certain subgroups (e.g., copies of free groups) into these full groups. The theorem specializes to the Higman–Thompson and Brin–Thompson families and links amenability of the orbit equivalence relation to non-strong ergodicity via the germ groupoid framework, providing a broad mechanism to study non-full crossed products in this setting.

Abstract

We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.
Paper Structure (6 sections, 7 theorems, 11 equations)

This paper contains 6 sections, 7 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groupoids and topological full groups
  4. Equivalence relations
  5. Strongly ergodic actions and full factors
  6. Main result

Key Result

Theorem 1

Let $V$ be the Thompson group and $V \curvearrowright X_2$ be the canonical action. Then the associated von Neumann algebra $L^{\infty}(X_2, \mu _2) \bar{\rtimes} V$ is non-amenable, non-full factor of type $\textrm{III}_{1/2}$.

Theorems & Definitions (16)

  • Theorem : see Corollary \ref{['HigmanThompsonalgebra']}
  • Theorem : see Theorem \ref{['main']}
  • Definition : DelarocheRenault
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition : see ZimmerConnesFeldmanWeissJacksonKechrisLouveau
  • Definition : Connes
  • Lemma 3.1
  • ...and 6 more