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

We show that for a finitely generated group $G$ and for every discrete character $\chi\colon G \rightarrow \mathbb{Z}$ 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 $FP_{n-1}$ containing the commutator subgroup $G'$ and such that $G/N$ is cyclic-by-finite, then $N$ is of homological type $FP_n$ and $G/N$ has finite virtual cohomological dimension $$vcd(G/N) = cd(G) - cd(N).$$ This completes the proof of the Rapaport Strasser conjecture that for a knot-like group $G$ with a finitely generated commutator subgroup $G'$ the commutator subgroup $G'$ is always free and generalises an earlier work by the author where the case when $G'$ is residually finite was proved. Another corollary is that a finitely presentable group $G$ with $def(G) > 0$ and such that $G'$ 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].