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Journal of Operator Theory

Volume 85, Issue 1, Winter 2021  pp. 217-228.

Von Neumann algebras of sofic groups with $\beta_{1}^{(2)}=0$ are strongly $1$-bounded

Authors: Dimitri Shlyakhtenko
Author institution: Department of Mathematics, UCLA, Los Angeles, CA 90095, U.S.A.

Summary: We show that if $\Gamma$ is a 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

Keywords: free probability, free entropy, $L^2$-Betti numbers

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