[Paper Review] Strong Rigidity of II$_1$ Factors Coming from Malleable Actions of Weakly Rigid Groups, I

Sorin Popa|arXiv (Cornell University)|May 21, 2003

Advanced Operator Algebra Research1 citations

TL;DR

This paper establishes strong rigidity for II$_1$ factors arising from malleable, mixing actions of weakly rigid groups on finite von Neumann algebras. By proving the uniqueness of the position of $L(G)$ inside the crossed product $M = N \times_\sigma G$, it computes the fundamental group $\mathcal{F}(M)$ in terms of shift weights for arithmetic groups like $\mathbb{Z}^2 \times SL(2,\mathbb{Z})$, resolving a longstanding problem by constructing II$_1$ factors with arbitrary countable fundamental groups.

ABSTRACT

We consider cross-product II$_1$ factors $M = N times_{\sigma} G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $\sigma: G o { ext{ m Aut}}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable'' and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=\Bbb Z^2 times SL(2, \Bbb Z)$. We deduce that for any countable group $S \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.

Motivation & Objective

• To establish strong rigidity for II$_1$ factors constructed as crossed products $M = N \times_\sigma G$ where $G$ is a discrete ICC group with infinite normal subgroups having relative property (T).
• To analyze the uniqueness of the subalgebra $L(G)$ inside $M$ under malleable and mixing actions $\sigma$ of $G$ on finite von Neumann algebras $N$.
• To compute the fundamental group $\mathcal{F}(M)$ for such factors, particularly for arithmetic groups like $\mathbb{Z}^2 \times SL(2,\mathbb{Z})$, using the weights of malleable shifts.
• To resolve a longstanding open problem of Murray and von Neumann by showing that for any countable subgroup $S \subset \mathbb{R}_+^*$, there exists a II$_1$ factor $M$ with $\mathcal{F}(M) = S$.

Proposed method

• Utilizes malleable actions $\sigma: G \to \text{Aut}(N)$ that are trace-preserving and mixing, with $G$ having infinite normal subgroups possessing relative property (T).
• Applies techniques from Popa's deformation/rigidity theory to analyze the position of $L(G)$ inside the crossed product factor $M = N \times_\sigma G$.
• Employs the malleability of the action to control asymptotic behavior and derive rigidity of the inclusion $L(G) \subset M$.
• Uses the structure of weighted Bernoulli and Bogoliubov shifts as concrete examples of malleable actions to compute the fundamental group via weight parameters.
• Relies on the uniqueness of the Cartan subalgebra and the spectral gap induced by relative property (T) to constrain possible embeddings of $L(G)$.
• Applies results on the fundamental group of crossed products to show that $\mathcal{F}(M)$ is determined by the weights of the shift, especially in arithmetic group settings.

Experimental results

Research questions

• RQ1 Under what conditions is the subalgebra $L(G)$ uniquely positioned inside the crossed product II$_1$ factor $M = N \times_\sigma G$ arising from a malleable, mixing action of $G$?
• RQ2 How does the fundamental group $\mathcal{F}(M)$ of such a crossed product factor depend on the weights of the malleable shift action?
• RQ3 Can the fundamental group of a II$_1$ factor be realized as any given countable subgroup $S \subset \mathbb{R}_+^*$, as conjectured by Murray and von Neumann?
• RQ4 What role does relative property (T) of an infinite normal subgroup of $G$ play in the rigidity of the inclusion $L(G) \subset M$?
• RQ5 To what extent do malleable actions of weakly rigid groups constrain the structure of the resulting II$_1$ factors?

Key findings

• The position of $L(G)$ inside the crossed product factor $M = N \times_\sigma G$ is uniquely determined when $G$ is an ICC group with an infinite normal subgroup having relative property (T), and $\sigma$ is a malleable, mixing action.
• For arithmetic groups such as $G = \mathbb{Z}^2 \times SL(2,\mathbb{Z})$, the fundamental group $\mathcal{F}(M)$ is computed explicitly in terms of the weights of the malleable shift action.
• The fundamental group $\mathcal{F}(M)$ of the constructed II$_1$ factors can be any countable subgroup $S \subset \mathbb{R}_+^*$, thus solving a long-standing problem of Murray and von Neumann.
• The malleability of the action is essential in controlling the asymptotic behavior and ensuring the uniqueness of $L(G)$ in $M$, even when the action is not rigid in the classical sense.
• The rigidity result holds despite the absence of full rigidity in the group action, due to the interplay between relative property (T) and malleability.
• The construction provides explicit examples of II$_1$ factors with prescribed fundamental groups, demonstrating the flexibility of the deformation/rigidity framework in von Neumann algebra theory.