  We prove that the rational number |n/m| is an invariant of the group von
Neumann algebra of the Baumslag-Solitar group BS(n,m). More precisely, if
L(BS(n,m)) is isomorphic with L(\BS(n',m')), then |n'/m'| = |n/m| or |m/n|. We
obtain this result by associating to abelian, but not maximal abelian,
subalgebras of a II_1 factor, an equivalence relation that can be of type III.
In particular, we associate to L(BS(n,m)) a canonical equivalence relation of
type III_|n/m|.
