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

We show that if $Γ$ is an infinite finitely generated finitely presented sofic group with zero first $L^{2}$ Betti number then the von Neumann algebra $L(Γ)$ is strongly $1$-bounded in the sense of Jung. In particular, $L(Γ)\not\cong L(Λ)$ if $Λ$ 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.

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