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Von Neumann Algebras of Sofic Groups with $β_{1}^{(2)}=0$ are Strongly $1$-Bounded (1604.08606v2)
Published 28 Apr 2016 in math.OA
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.