The zero divisor conjecture for some solvable groups

Abstract

Let F be a field and G a group. The zero divisor conjecture states that if G is torsion free, then the group algebra F[G] is torsion free. A series of papers by various authors have resulted in a proof of this conjecture for polycyclic-by-finite groups. The next most natural step would seem to be groups which are poly-(torsion free rank one abelian)-by-finite. These are pcecisely the solvable groups of finite cohomological dimension. A perhaps more attractive description of these groups is the solvable-by-finite subgroups of GLn(Q), Q being the rational numbers. We are able to prove this conjecture for the class of these groups where the primes in the finite top are different from the primes that make the rank one abelian factors non finitely generated. The key ingredient in the proof is a localization theorem which makes these non-Noetherian group rings Noetherian. If A is a finite rank torson free abelian group, pick a free abelian subgroup F such that A/F is torsion. The spectrum (spec A) is the set of primes p such that the p-primary component of A/F is infinite [14, p. 167]. If N is a torsion free finite rank nilpotent group, Spec (N) is defined to be the union of the spectrum of the factors of the lower central series. We prove the