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\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].