  We give new information about the relationship between the low-dimensional
homology of a group and its derived series. This yields information about how
the low-dimensional homology of a topological space constrains its fundamental
group. Applications are given to detecting when a set of elements of a group
generates a subgroup ``large enough'' to map onto a non-abelian free solvable
group, and to concordance and grope cobordism of links. We also greatly
generalize several key homological results employed in recent work of
Cochran-Orr-Teichner, in the context of classical knot concordance.
  In 1963 J. Stallings established a strong relationship between the
low-dimensional homology of a group and its lower central series quotients. In
1975 W. Dwyer extended Stallings' theorem by weakening the hypothesis on the
second homology groups. The naive analogues of these theorems for the derived
series are false. In 2003 the second author introduced a new characteristic
series, associated to the derived series, called the torsion-free derived
series. The authors previously established a precise analogue, for the
torsion-free derived series, of Stallings' theorem. Here our main result is the
analogue of Dwyer's theorem for the torsion-free derived series. We also prove
a version of Dwyer's theorem for the rational lower central series. We apply
these to give new results on the Cochran-Orr-Teichner filtration of the
classical link concordance group.
