Vanishing first cohomology and strong 1-boundedness for von Neumann algebras
arXiv:2110.12324
Abstract
In this paper, we obtain a new proof result of Shlyakhtenko which states that if is a sofic, finitely presented group with vanishing first -Betti number, then 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 provide a short proof using works of Shlyakhtenko and Shalom 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.
17 pages; split off from arXiv:2107.03278. This is the final version to appear as such in Journal of Noncommutative Geometry