Strong rigidity of \(\text{II}_1\) factors arising from malleable actions of \(w\)-rigid groups. II
type: Item
instance of: scholarly article
title: Strong rigidity of \(\text{II}_1\) factors arising from malleable actions of \(w\)-rigid groups. II
zbMATH Open document ID: 1120.46044
published in: Inventiones Mathematicae
publication date: 2006-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0407103
review text: The paper expands its author's programme on von Neumann algebra rigidity, which has recently led to spectacular results on the fundamental group of II\(_1\) factors [Ann. Math. (2) 163, 809--899 (2006; Zbl 1120.46045), part I of the paper, Invent. Math. 165, 369--408 (2006; Zbl 1120.46043). Isomorphisms between group measure space II\(_1\) factors \(L^\infty (X,\mu) \rtimes_\sigma G\) arising from free ergodic measure-preserving actions of discrete countable groups on probability spaces are considered this time. The typical assumption on the group \(G\) is to be either \textit{ICC} (infinite conjugacy class) and \(w\)-rigid (i.e., to contain an infinite normal subgroup \(H\) with relative Kazhdan-Margulis property T), or to belong to the class \(w{\mathcal T}_0\) of groups with infinite subgroups \(H\) with relative property T and such that \(H\) is weakly quasi normal in \(G\) and is not virtually Abelian. The typical assumption on the action \(\sigma\) is to be sub \(s\)-malleable (as defined in part I and clustering (a strong notion of mixing defined in the paper). Bernoulli actions of ICC groups \(G\), acting on infinite product probability spaces \((X,\mu)=\prod_{g\in G} (X_0,\mu_0)\) by \(\sigma_{g_0} ((x_g)_g)=(x_{g_0^{-1} g})_g\), provide examples of sub \(s\)-malleable clustering actions. Measure group II\(_1\) factors are shown to exhibit very strong rigidity properties under these kinds of assumptions, in the sense that any isomorphism between \(M_1 =L^\infty (X_1,\mu_1)\rtimes_{\sigma_1} G_1\) and \(L^\infty (X_2,\mu_2) \rtimes_{\sigma_2} G_2\) is essentially implemented by a probability space isomorphism \((X_1,\mu_1)\cong (X_2,\mu_2)\) which conjugates the actions \(\sigma_1\) and \(\sigma_2\) with respect to a group isomorphism \(G_1 \cong G_2\). In fact, the following more general result is proved. Theorem. Let \(G_i\) be a countable discrete group acting freely and ergodically by \(\sigma_i\) on a standard probability space \((X_i,\mu_i)\) and denot…
reviewed by: Florin P. Boca
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 22F10
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 37A20
zbMATH DE Number: 5051180
zbMATH Keywords: II\(_1\) factors
zbMATH Keywords: group measure space factors
zbMATH Keywords: \(w\)-rigid groups
zbMATH Keywords: \(w{\mathcal T}_0\) groups
zbMATH Keywords: malleable clustering actions
zbMATH Keywords: Bernoulli actions
zbMATH Keywords: rigidity
zbMATH Keywords: von Neumann equivalence strong rigidity
zbMATH Keywords: orbit equivalence strong rigidity
zbMATH Keywords: conjugacy of actions
author: Sorin Popa
MaRDI profile type: Publication
OpenAlex ID: W4233659647
arXiv ID: math/0407103
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DOI: 10.1007/S00222-006-0502-3
description: scientific article