Classification of Invariant Subalgebras in a class of factors with property (T)

TL;DR

The paper classifies all $G_n$-invariant von Neumann subalgebras of the group factor $L(G_n)$ for $G_n=\mathbb{Z}^n\rtimes SL_n(\mathbb{Z})$, showing they arise only from normal subgroups or from invariant subalgebras associated with the semi-direct product structure. Using a character-analytic approach tied to $SL_n(\mathbb{Z})$-invariant measures on $\mathbb{Z}^n$, the authors treat odd and even $n$ separately due to differences in $SL_n(\mathbb{Z})$ centers, establishing that in odd $n$ the invariant subalgebras are either $L(H)$ or $A_d$, and in even $n$ the list expands to include $L(\mathbb{Z}^n\rtimes \{\pm I_n\})$ and $L(2\mathbb{Z}^n\rtimes \{\pm I_n\})$ alongside the abelian cases. A corollary asserts the existence of a unique maximal Haagerup $G_n$-invariant subalgebra, providing a partial rigidity picture for property (T) groups without ISR. The methods extend and refine previous work for $n=2$, and the results contribute to understanding invariant subalgebras in property (T) group factors and their Haagerup radicals.

Abstract

Let $n\geq 2$ and $G_n=\mathbb{Z}^n\rtimes SL_n(\mathbb{Z})$. We classify all $G_n$-invariant von Neumann subalgebras in $L(G_n)$. For $n=2$, this gives an alternative proof of the previous result of Jiang-Liu. For $n\geq 3$, this gives the first class of property (T) groups without the invariant subalgebras rigidity property but invariant subalgebras in the corresponding group factors can still be classified. As a corollary, $L(G_n)$ admits a unique maximal Haagerup $G_n$-invariant von Neumann subalgebra.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof