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 precisely 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.
