Remarks on the diagonal embedding and strong 1-boundedness

TL;DR

This paper investigates the properties of von Neumann algebras arising from certain hyperbolic groups, showing many are not strongly 1-bounded and analyzing subalgebras of diagonal embeddings.

Contribution

It identifies a broad class of hyperbolic groups with von Neumann algebras that lack strong 1-boundedness and examines properties of intermediate subalgebras in diagonal embeddings.

Findings

Hyperbolic groups over $F_2$ have von Neumann algebras not strongly 1-bounded.

Intermediate subalgebras of the diagonal embedding of $L(F_2)$ lack Property (T).

The results extend understanding of the structure of von Neumann algebras from hyperbolic groups.

Abstract

We identify a large class of hyperbolic groups whose von Neumann algebras are not strongly 1-bounded: Sela's hyperbolic towers over $F_2$ subgroups. We also show that any intermediate subalgebra of the diagonal embedding of $L(F_2)$ into its ultrapower doesn't have Property (T).