Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

TL;DR

This paper provides a simplified proof that certain von Neumann algebras associated with sofic groups with vanishing first - Betti number are strongly 1-bounded, extending to groups with Property T.

Contribution

It offers a new, simplified proof of strong 1-boundedness for von Neumann algebras of specific sofic groups, improving on previous technical methods.

Findings

Von Neumann algebras of sofic groups with vanishing first -Betti number are strongly 1-bounded.

Von Neumann algebras of sofic groups with Property T are strongly 1-bounded.

The proof simplifies previous approaches by adapting Jung's iterative estimate technique.

Abstract

In this paper, we obtain a new proof result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $\ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede-Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also provide a short proof using works of Shlyakhtenko and Shalom that the von Neumann algebras of sofic groups with Property T are strongly 1 bounded, which is a special case of another result by the authors.