# When can the group von Neumann algebra of a one-relator group be isomorphic to a free group factor?

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.)

• Following the $\ell^2$-Betti numerology, this should never happen if $G$ is torsionfree (unless $G$ is abelian). Linnell and Dicks showed that the first $\ell^2$-Betti number of a torsionfree 2-generator 1-relator group vanishes. If it where isomorphic to an interpolated free group factor $L\mathbb F_t$, then one would expect that $t=1$ (being equal to the first $\ell^2$-Betti number plus $1$. – Andreas Thom Dec 7 '10 at 18:10
• To go along with what Andreas wrote, if you allow torsion then I believe it was Dykema and Radulescu who showed that the groups $\langle a, b \ | \ b^k \rangle$ for $2 \leq k < \infty$ always give interpolated free group factors $L\mathbb F_t$, with $1 < t < 2$. – Jesse Peterson Dec 7 '10 at 21:35
• Jesse has answered this question as asked. Thanks! (I should have looked more carefully for such results before asking the question!!!) – Jon Bannon Dec 7 '10 at 22:31
• I've changed the question a bit to allow for more thoughts. – Jon Bannon Dec 7 '10 at 22:37
• Also, do you mean free entropy when you say free entropy dimension above? The free entropy dimension is always finite when the factor embeds into $R^\omega$. – Jesse Peterson Dec 7 '10 at 23:15