We show that for a finitely generated group and for every discrete character any matrix ring over the Novikov ring is von Neumann finite. As a corollary we obtain that if is a non-trivial discrete group with a finite CW-complex of dimension and Euler characteristics zero and is a normal subgroup of of type containing the commutator subgroup and such that is cyclic-by-finite, then is of homological type and has finite virtual cohomological dimension
This completes the proof of the Rapaport Strasser conjecture that for a knot-like group with a finitely generated commutator subgroup the commutator subgroup is always free and generalises an earlier work by the author where the case when is residually finite was proved. Another corollary is that a finitely presentable group with and such that is finitely generated and perfect can be only or , a result conjectured by A. J. Berrick and J. Hillman in .
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. 931–943DOI 10.4171/CMH/81