  In this paper, we obtain a new proof result of Shlyakhtenko 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
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.
