Von Neumann Algebras of Sofic Groups with ${\beta }_1^{(2)}=0$ are Strongly $1$-Bounded

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, $L(\Gamma )\ncong L(\Lambda )$ 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.

Fixed a few typos and (thanks to improved results by B. Hayes) dropped the condition that groups contain an element of infinite order

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