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Examples of W$^*$ and C$^*$-superrigid product groups

Jakub Curda, Daniel Drimbe

TL;DR

The paper advances the understanding of rigidity phenomena for product groups by introducing the class $\mathcal{C}_{AFP}$ of amalgamated free product groups and proving a robust product rigidity result: if $G_1,\dots,G_n\in\mathcal{C}_{AFP}$ and $L(G_1\times\dots\times G_n)\cong L(H)$, then $H$ splits as a product with each factor's group von Neumann algebra matching up to stable isomorphism. It shows that products of groups from $\mathcal{C}_{AFP}$ are both $W^*$- and $C^*$-superrigid, and extends these rigidity results to include products with icc relatively solid factors, using Popa's intertwining-by-bimodules, relative amenability, and ultrapower techniques. A key part of the work identifies peripheral substructures inside the von Neumann algebra to recover the original product decomposition, enabling a full identification of the factors up to amplification. Collectively, these results provide new, non-amenable examples of product groups that are rigid in both the von Neumann algebraic and $C^*$-algebraic senses, with a clear pathway from operator-algebraic data to the underlying group structure.

Abstract

We provide a new large class $\mathcal C_{AFP}$ of amalgamated free product groups for which the product rigidity result from [CdSS15] holds: if $G_1,\dots,G_n\in\mathcal C_{AFP}$ and $H$ is any group such that $L(G_1\times\dots\times G_n)\cong L(H)$, then there exists a product decomposition $H=H_1\times\dots\times H_n$ such that $L(H_i)$ is stably isomorphic to $L(G_i)$, for any $1\leq i\leq n$. The class $\mathcal C_{AFP}$ contains $W^*$ and $C^*$-superrigid groups from [CD-AD20]. Consequently, we obtain examples of product groups that are both $W^*$ and $C^*$-superrigid.

Examples of W$^*$ and C$^*$-superrigid product groups

TL;DR

The paper advances the understanding of rigidity phenomena for product groups by introducing the class of amalgamated free product groups and proving a robust product rigidity result: if and , then splits as a product with each factor's group von Neumann algebra matching up to stable isomorphism. It shows that products of groups from are both - and -superrigid, and extends these rigidity results to include products with icc relatively solid factors, using Popa's intertwining-by-bimodules, relative amenability, and ultrapower techniques. A key part of the work identifies peripheral substructures inside the von Neumann algebra to recover the original product decomposition, enabling a full identification of the factors up to amplification. Collectively, these results provide new, non-amenable examples of product groups that are rigid in both the von Neumann algebraic and -algebraic senses, with a clear pathway from operator-algebraic data to the underlying group structure.

Abstract

We provide a new large class of amalgamated free product groups for which the product rigidity result from [CdSS15] holds: if and is any group such that , then there exists a product decomposition such that is stably isomorphic to , for any . The class contains and -superrigid groups from [CD-AD20]. Consequently, we obtain examples of product groups that are both and -superrigid.
Paper Structure (15 sections, 20 theorems, 12 equations)

This paper contains 15 sections, 20 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology
  4. Intertwining-by-bimodules
  5. Relative amenability
  6. Quasinormalizers of groups and von Neumann algebras
  7. Relatively solid groups
  8. Amalgamated free product groups and class $\mathcal{C}_{AFP}$
  9. Solidity type results
  10. Product rigidity for groups in $\mathcal{C}_{AFP}$ and relatively solid groups
  11. Identification of peripheral structure
  12. Proofs of the main results
  13. Proof of Theorem \ref{['B']}
  14. Proof of Theorem \ref{['A']}
  15. Proof of Corollary \ref{['corA']}

Key Result

Theorem A

Let $G_1,\dots,G_n$ be groups from $\mathcal{C}^0_{AFP}$ and denote $G=G_1\times\dots\times G_n$. Let $H$ be an arbitrary group and let $\theta: L(G)^t\to L(H)$ be a $*$-isomorphism for some $t>0$. Then $t=1$ and $G\cong H$. Moreover, there exist a group isomorphism $\delta: G\to H$, a unitary $w\in

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem A
  • Corollary B
  • Example 1.2
  • Theorem C
  • Theorem 2.1: Po03
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Po03FGS10
  • Lemma 2.5: FGS10
  • ...and 14 more