Let $G=\langle a,b | R \rangle$ be a one-relator group. When can the left group von Neumann algebra $LG$ be isomorphic to a free group factor? Jesse and Andreas have "trapped the lion" pretty well with their comments below.

A bit more modest related question: if $L_{a}$ and $L_{b}$ are the unitary elements in $LG$ corresponding to $a$ and $b$, respectively, can the free entropy of ($L_{a}+L_{a}^{*},L_{b}+L_{b}^{*}$) be finite? For the definition of free entropy, see Voiculescu's survey: http://arxiv.org/PS_cache/math/pdf/0103/0103168v1.pdf.

(It is possible for a set of generators of a type $II_{1}$-factor that is not a free group factor to have finite free entropy. Nate Brown establishes this in http://arxiv.org/abs/math/0403294.)