JournalscmhVol. 81 , No. 4DOI 10.4171/cmh/81

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

  • Dessislava H. Kochloukova

    IMECC - UNICAMP, Campinas, Sp, Brazil
Some Novikov rings that are von Neumann finite and knot-like groups cover

Abstract

We show that for a finitely generated group GG and for every discrete character χ ⁣:GZ\chi\colon G \rightarrow \mathbb{Z} any matrix ring over the Novikov ring ZG^χ\widehat{\mathbb{Z}G}_{\chi} is von Neumann finite. As a corollary we obtain that if GG is a non-trivial discrete group with a finite K(G,1)K(G,1) CW-complex YY of dimension nn and Euler characteristics zero and NN is a normal subgroup of GG of type FPn1FP_{n-1} containing the commutator subgroup GG' and such that G/NG/N is cyclic-by-finite, then NN is of homological type FPnFP_n and G/NG/N has finite virtual cohomological dimension

vcd(G/N)=cd(G)cd(N).vcd(G/N) = cd(G) - cd(N).

This completes the proof of the Rapaport Strasser conjecture that for a knot-like group GG with a finitely generated commutator subgroup GG' the commutator subgroup GG' is always free and generalises an earlier work by the author where the case when GG' is residually finite was proved. Another corollary is that a finitely presentable group GG with def(G)>0def(G) > 0 and such that GG' is finitely generated and perfect can be only Z\mathbb{Z} or Z2\mathbb{Z}^2, a result conjectured by A. J. Berrick and J. Hillman in [1].