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McDuff superrigidity for group II$_1$ factors

Juan Felipe Ariza Mejía, Ionuţ Chifan, Denis Osin, Bin Sun

TL;DR

The paper identifies the first examples of ICC groups for which the group von Neumann algebra is McDuff and exhibits McDuff superrigidity, showing that any H with L(H) ≅ L(G) must split as G × A with A ICC amenable. It develops a novel integration of group-theoretic Dehn filling and wreath-like products with uniformly bounded cocycles, constructs arrays to detect cocycle support, and leverages Gaussian and Ioana deformations alongside Popa intertwining to reconstruct infinite direct sums from W*-equivalence. The combination of these techniques yields a continuum of non-isomorphic McDuff superrigid groups built from property (T) wreath-like product summands and demonstrates strong rigidity phenomena at the von Neumann algebra level. The results have significant implications for the classification of II1 factors arising from complex group constructions and illustrate a deep interaction between geometric group theory and operator algebra rigidity.

Abstract

Developing new techniques at the interface of geometric group theory and von Neumann algebras, we identify the first examples of ICC groups $G$ whose von Neumann algebras are McDuff and exhibit a new rigidity phenomenon, termed McDuff superrigidity: an arbitrary group $H$ satisfying $\mathcal{L}(H)\cong \mathcal{L}(G)$ decomposes as $H \cong G\times A$ for an ICC amenable group $A$.

McDuff superrigidity for group II$_1$ factors

TL;DR

The paper identifies the first examples of ICC groups for which the group von Neumann algebra is McDuff and exhibits McDuff superrigidity, showing that any H with L(H) ≅ L(G) must split as G × A with A ICC amenable. It develops a novel integration of group-theoretic Dehn filling and wreath-like products with uniformly bounded cocycles, constructs arrays to detect cocycle support, and leverages Gaussian and Ioana deformations alongside Popa intertwining to reconstruct infinite direct sums from W*-equivalence. The combination of these techniques yields a continuum of non-isomorphic McDuff superrigid groups built from property (T) wreath-like product summands and demonstrates strong rigidity phenomena at the von Neumann algebra level. The results have significant implications for the classification of II1 factors arising from complex group constructions and illustrate a deep interaction between geometric group theory and operator algebra rigidity.

Abstract

Developing new techniques at the interface of geometric group theory and von Neumann algebras, we identify the first examples of ICC groups whose von Neumann algebras are McDuff and exhibit a new rigidity phenomenon, termed McDuff superrigidity: an arbitrary group satisfying decomposes as for an ICC amenable group .

Paper Structure

This paper contains 28 sections, 43 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Outline of the paper
  4. 2.1. Group theoretic Dehn filling and wreath-like products with bounded cocycle.
  5. 2.2. Arrays and $W^*$-rigidity of infinite products.
  6. Group-theoretic preliminaries
  7. Notation
  8. Van Kampen diagrams
  9. Hyperbolically embedded subgroups
  10. Isolated components and geodesic polygons
  11. Separating cosets
  12. Diagram surgery
  13. Sufficiently long Dehn filling
  14. Cohen--Lyndon triples
  15. Wreath-like product groups with $2$-cocycles of uniformly bounded support
  16. ...and 13 more sections

Key Result

Theorem 1.3

There exists a family of countable $W^\ast$-McDuff groups $\{ G_i\}_{i\in I}$ such that $|I|=2^{\aleph_0}$, $G_i\not\cong G_j$ whenever $i\ne j$, and every $G_i$ is McDuff superrigid.

Figures (2)

  • Figure 1: Case 2
  • Figure 2: Case 3

Theorems & Definitions (99)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Definition 3.1
  • ...and 89 more