Lamplighter groups and von Neumann‘s continuous regular ring

Abstract:

Let $\Gamma$ be a discrete group. Following Linnell and Schick one can define a continuous ring $c(\Gamma )$ associated with $\Gamma$. They proved that if the Atiyah Conjecture holds for a torsion-free group $\Gamma$, then $c(\Gamma )$ is a skew field. Also, if $\Gamma$ has torsion and the Strong Atiyah Conjecture holds for $\Gamma$, then $c(\Gamma )$ is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group $\Gamma =\mathbb {Z}_2\wr \mathbb {Z}$. It is known that $\mathbb {C}(\mathbb {Z}_2\wr \mathbb {Z})$ does not even have a classical ring of quotients. Our main result is that if $H$ is amenable, then $c(\mathbb {Z}_2\wr H)$ is isomorphic to a continuous ring constructed by John von Neumann in the 1930s.