I would like to understand if there is a discrete infinite group of exponential growth/intermediate growth such that its group von Neumann algebra is a $II\_1$ factor. I would be happy to get an explicit example if such group exists. Many thanks.



an example of discrete factor group of exponential growth
(For non-specialists: $G$ is embedded into $\mathcal{B}(\ell^2\_{\mathbf{C}}(G))$ as unitaries induced by left translations, and its von Neumann algebra $LG$ is its bicommutant. To say that $LG$ is a factor means that the center of $LG$ is reduced to scalars.)


By a classical result of Murray and von Neumann, the von Neumann algebra of a group $G$ is a factor (and then $II\_1$-factor) iff the group is ICC, that is, $\mathrm{FC}(G)=1$, where $\mathrm{FC}(G)$ is the union of finite conjugacy classes of $G$. Equivalently this means that $G$ has trivial center and trivial finite radical, i.e., has no nontrivial finite normal subgroup. (That this condition is necessary is trivial.)


If $G\neq 1$ is finitely generated and ICC, then $G$ is not virtually nilpotent. Among other groups, there are plenty of ICC groups: non-abelian free groups, etc, and also many elementary amenable ones too (e.g. polycyclic). In intermediate growth there are plenty of examples too (just-infinite groups such as Grigorchuk groups...).


Actually, any non-virtually-nilpotent finitely generated group has a nontrivial ICC quotient.


