Narrow Results

Let G = ℤ/2ℤ ≀ ℤ be the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1 – x is invertible, with x a generator of ℤ ⊂ G. Then R is not flat over kG. If k = ℂ, this applies in particular to the algebra of unbounded operators affiliated to the group von Neumann algebra of G.

We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.