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

We show that if $\mathrm{\Gamma }$ is an infinite finitely generated finitely presented sofic group with zero first $L^2$ Betti number then the von Neumann algebra $L(\mathrm{\Gamma })$ is strongly $1$-bounded in the sense of Jung. In particular, $L(\mathrm{\Gamma })\cong ̸L(\mathrm{\Lambda })$ if $\mathrm{\Lambda }$ 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.