In 1987 Bieri, Neumann, and Strebel introduced a geometric invariant of a discrete group G. Among other things, the BNS-invariant determines which subgroups containing the commutator subgroup are finitely generated. This talk with be something of an introduction to geometric group theory, focused on presenting enough background to make the description of the BNS-invariant of the pure braid group comprehensible to a broad audience. The main result is joint work with Nic Koban and Jon McCammond.a**