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

### Dessislava H. Kochloukova

IMECC - UNICAMP, Campinas, Sp, Brazil

## Abstract

We show that for a finitely generated group $G$ and for every discrete character $χ:G→Z$ any matrix ring over the Novikov ring $ZG_{χ}$ 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

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 $Z$ or $Z_{2}$, a result conjectured by A. J. Berrick and J. Hillman in [1].

## Cite this article

Dessislava H. Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups. Comment. Math. Helv. 81 (2006), no. 4, pp. 931–943

DOI 10.4171/CMH/81