Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

W$^*$-superrigidity for property (T) groups with infinite center

Ionuţ Chifan, Adriana Fernández Quero, Denis Osin, Hui Tan

TL;DR

This work advances Connes' rigidity program for property (T) groups with infinite center by constructing and analyzing generalized wreath-like product groups and their central extensions. It proves virtual and, under favorable hypotheses, strong W$^*$-rigidity theorems for twisted group factors, showing that centers and fiber structures can be reconstructed from von Neumann algebras. A key innovation is a quotienting technique that yields nonsplit central extensions with controlled centers and Out$(ullet)$, enabling a broad class of W$^*$-superrigid property (T) groups with infinite centers. The results connect to and extend recent frameworks for twisted group von Neumann algebras, and they pave the way for refined conjectures about Connes' rigidity in the central-extension setting, including open problems on the existence of flexible groups and further fiberwise rigidity phenomena.

Abstract

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W$^*$-superrigid property (T) group with infinite center. In the course of proving our main results, we also generalize the main W$^*$-superrigidity result from \cite{cios22} to twisted group factors.

W$^*$-superrigidity for property (T) groups with infinite center

TL;DR

This work advances Connes' rigidity program for property (T) groups with infinite center by constructing and analyzing generalized wreath-like product groups and their central extensions. It proves virtual and, under favorable hypotheses, strong W-rigidity theorems for twisted group factors, showing that centers and fiber structures can be reconstructed from von Neumann algebras. A key innovation is a quotienting technique that yields nonsplit central extensions with controlled centers and Out, enabling a broad class of W-superrigid property (T) groups with infinite centers. The results connect to and extend recent frameworks for twisted group von Neumann algebras, and they pave the way for refined conjectures about Connes' rigidity in the central-extension setting, including open problems on the existence of flexible groups and further fiberwise rigidity phenomena.

Abstract

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W-superrigid property (T) group with infinite center. In the course of proving our main results, we also generalize the main W-superrigidity result from \cite{cios22} to twisted group factors.

Paper Structure

This paper contains 12 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Generalized wreath-like product groups
  4. Generalized wreath-like products
  5. Central extensions of generalized wreath-like product groups
  6. Preliminaries on von Neumann algebras
  7. Intertwining techniques
  8. Twisted group von Neumann algebras
  9. Direct integral decompositions
  10. Proof of \ref{['strongsuperrig']}
  11. Rigidity for property (T) central extensions
  12. A broader open problem

Key Result

Theorem 3.3

Let $G$ be a group and let $\{H_\lambda\}_{\lambda\in \Lambda} \hookrightarrow^{h} (G,X)$ be a family of hyperbolically embedded subgroups for some $X \subset G$. Then for every $\lambda\in \Lambda$ one can find a finite subset $F_\lambda \subset H_\lambda\setminus \{1\}$ such that for every normal

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Sun
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • Theorem 3.7
  • proof
  • ...and 48 more