Aimed at geometric applications, we prove the homology cobordism invariance
of the $L^2$-betti numbers and $L^2$-signature defects associated to the class
of amenable groups lying in Strebel's class $D(R)$, which includes some
interesting infinite/finitenon-torsion-free groups. The proofs include the only
prior known condition, that $\Gamma$ is a poly-torsion-free abelian group (or
potentially a finite $p$-group.) We define a new commutator-type series which
refines Harvey's torsion-free derived series of groups, using the localizations
of groups and rings of Bousfield, Vogel, and Cohn.