Von Neumann Algebras of Sofic Groups with $\beta_{1}^{(2)}=0$ are Strongly $1$-Bounded

TL;DR

This paper proves that for certain sofic groups with zero first $L^{2}$ Betti number, their von Neumann algebras are strongly 1-bounded, distinguishing them from free group factors with higher free entropy dimension.

Contribution

It establishes strong 1-boundedness of von Neumann algebras of specific sofic groups using non-microstates entropy methods, providing new insights into their structure.

Findings

Von Neumann algebras of certain sofic groups are strongly 1-bounded.

Such algebras are not isomorphic to free group factors with higher free entropy.

A new proof technique for Jung's estimate using non-microstates entropy.

Abstract

We show that if $\Gamma$ is an infinite finitely generated finitely presented sofic group with zero first $L^{2}$ Betti number then the von Neumann algebra $L(\Gamma)$ is strongly $1$-bounded in the sense of Jung. In particular, $L(\Gamma)\not\cong L(\Lambda)$ if $\Lambda$ is any group with free entropy dimension $>1$, for example a free group. The key technical result is a short proof of an estimate of Jung using non-microstates entropy techniques.