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
DOI: http://dx.doi.org/10.7900/jot.2019oct21.2270
Keywords: free probability, free entropy, $L^2$-Betti numbers
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