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

D. Shlyakhtenko

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arXiv · 1604.08606

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

D. Shlyakhtenko

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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.

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BibTeX RIS

D. Shlyakhtenko. 2016-04-28. Von Neumann Algebras of Sofic Groups with $\beta_{1}^{(2)}=0$ are Strongly $1$-Bounded. https://arxiv.org/abs/1604.08606

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