Some Novikov rings that are von Neumann finite and knot-like groups

Abstract

We show that for a finitely generated group $G$ and for every discrete character $\chi :G\to \mathbb{Z}$ any matrix ring over the Novikov ring ${\widehat{\mathbb{Z}G}}_{\chi }$ is von Neumann finite. As a corollary we obtain that if $G$ is a non-trivial discrete group with a finite $K(G,1)$ CW-complex $Y$ of dimension $n$ and Euler characteristics zero and $N$ is a normal subgroup of $G$ of type $F{P}_{n-1}$ containing the commutator subgroup $G^{\prime }$ and such that $G/N$ is cyclic-by-finite, then $N$ is of homological type $F{P}_n$ and $G/N$ has finite virtual cohomological dimension

This completes the proof of the Rapaport Strasser conjecture that for a knot-like group $G$ with a finitely generated commutator subgroup $G^{\prime }$ the commutator subgroup $G^{\prime }$ is always free and generalises an earlier work by the author where the case when $G^{\prime }$ is residually finite was proved. Another corollary is that a finitely presentable group $G$ with $def(G)>0$ and such that $G^{\prime }$ is finitely generated and perfect can be only $\mathbb{Z}$ or ${\mathbb{Z}}^2$, a result conjectured by A. J. Berrick and J. Hillman in [1].