Lamplighter groups and von Neumann's continuous regular rings

TL;DR

This paper explores the structure of continuous rings associated with lamplighter groups, showing that for amenable groups, these rings are isomorphic to von Neumann's classical continuous rings, extending understanding beyond the Atiyah conjecture.

Contribution

It demonstrates that for lamplighter groups with amenable base groups, the associated continuous ring is isomorphic to von Neumann's classical continuous ring, providing new insights into their algebraic structure.

Findings

For amenable groups, the continuous ring c(Z_2 wr H) is isomorphic to von Neumann's continuous ring.

The paper extends the understanding of continuous rings beyond cases where the Atiyah Conjecture holds.

It shows that c(Z_2 wr H) does not have a classical ring of quotients when the Strong Atiyah Conjecture fails.

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 $\Gamma=Z_2\wr Z$. It is known that $C(Z_2\wr Z)$ does not even have a classical ring of quotients. Our main result is that if $H$ is amenable, then $c(Z_2\wr H)$ is isomorphic to a continuous ring constructed by John von Neumann in the $1930's$.