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

D. Shlyakhtenko

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

Operator Algebras 2016-09-16 v2

Authors: D. Shlyakhtenko

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.

Keywords

sofic groups von Neumann algebras L^2 Betti number strongly 1-bounded free entropy dimension non-microstates entropy

Cite

@article{arxiv.1604.08606,
  title  = {Von Neumann Algebras of Sofic Groups with $\beta_{1}^{(2)}=0$ are Strongly $1$-Bounded},
  author = {D. Shlyakhtenko},
  journal= {arXiv preprint arXiv:1604.08606},
  year   = {2016}
}

Comments

Fixed a few typos and (thanks to improved results by B. Hayes) dropped the condition that groups contain an element of infinite order

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

Abstract

Keywords

Cite

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