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

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.