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A character approach to the ISR property

Artem Dudko, Yongle Jiang

TL;DR

The paper develops a character-centric framework to study invariant von Neumann subalgebras rigidity (ISR) and introduces the non-factorizable regular character property (NFRC) as a sufficient condition for ISR in ICC groups with trivial amenable radical. It shows that NFRC implies ISR and applies indecomposable-character classifications to establish ISR for several classes, including approximately finite (AF) groups, and constructs non-amenable groups with ISR even when the amenable radical is nontrivial. Beyond NFRC, a modified character approach yields ISR for broader constructions such as products of character-rigid groups and certain generalized wreath products, expanding the gallery of ISR groups. The work also provides explicit counterexamples and open questions, clarifying the limits of NFRC and guiding future exploration into ISR for amenable and acylindrically hyperbolic groups, Bratteli AF groups, and topological full groups. Overall, the paper advances a unifying character-based methodology for ISR and highlights both its reach and its boundaries in group von Neumann algebra rigidity.

Abstract

We develop a character approach to study the invariant von Neumann subalgebras rigidity property (abbreviated as the ISR property) introduced in Amrutam-Jiang's work. First, we introduce the non-factorizable regular character property for groups and show that this implies the ISR property for any infinite ICC groups with trivial amenable radical.Various examples are shown to have this property. Second, we apply known classification results on indecomposable characters to show approximately finite groups have the ISR property. Based on this approach, we also construct non-amenable groups with the ISR property while having non-trivial amenable radical or without the non-factorizable regular character property.

A character approach to the ISR property

TL;DR

The paper develops a character-centric framework to study invariant von Neumann subalgebras rigidity (ISR) and introduces the non-factorizable regular character property (NFRC) as a sufficient condition for ISR in ICC groups with trivial amenable radical. It shows that NFRC implies ISR and applies indecomposable-character classifications to establish ISR for several classes, including approximately finite (AF) groups, and constructs non-amenable groups with ISR even when the amenable radical is nontrivial. Beyond NFRC, a modified character approach yields ISR for broader constructions such as products of character-rigid groups and certain generalized wreath products, expanding the gallery of ISR groups. The work also provides explicit counterexamples and open questions, clarifying the limits of NFRC and guiding future exploration into ISR for amenable and acylindrically hyperbolic groups, Bratteli AF groups, and topological full groups. Overall, the paper advances a unifying character-based methodology for ISR and highlights both its reach and its boundaries in group von Neumann algebra rigidity.

Abstract

We develop a character approach to study the invariant von Neumann subalgebras rigidity property (abbreviated as the ISR property) introduced in Amrutam-Jiang's work. First, we introduce the non-factorizable regular character property for groups and show that this implies the ISR property for any infinite ICC groups with trivial amenable radical.Various examples are shown to have this property. Second, we apply known classification results on indecomposable characters to show approximately finite groups have the ISR property. Based on this approach, we also construct non-amenable groups with the ISR property while having non-trivial amenable radical or without the non-factorizable regular character property.

Paper Structure

This paper contains 17 sections, 34 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Characters
  5. Group von Neumann algebras and the ISR property
  6. The non-factorizable regular character property
  7. Non-factorizable regular character for groups arising from infinite simple groups
  8. Non-factorizable regular characters for charfinite groups
  9. Examples without non-factorizable regular characters
  10. Questions on the non-factorizable regular character property
  11. Approximately finite groups
  12. Preliminaries on Bratteli diagrams and corresponding AF groups.
  13. Auxiliary lemmas.
  14. Groups with the ISR property from modified character approach
  15. Checking ISR for groups without non-factorizable regular characters
  16. ...and 2 more sections

Key Result

Theorem A

Let $G$ be an ICC group. If $G$ has non-factorizable regular character property, then every $G$-invariant subfactor of $L(G)$ is equal to $L(H)$ for some normal subgroup $H$ of $G$. If $G$ is further assumed to be non-amenable with trivial amenable radical, then $G$ has the ISR property.

Theorems & Definitions (77)

  • Theorem A: Theorem \ref{['theorem: NFRC consequences']}
  • Theorem B: Theorem \ref{['prop: example of nonamenable groups with ISR but nontrivial amenable radical']}
  • Theorem C: Theorem \ref{['thm: approximately finite groups have the ISR property']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 67 more