Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

Abstract

We obtain a new proof of Shlyakhtenko's result which states that if $G$ is a sofic, finitely presented group with vanishing first ${\ell }^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.