Gretchen Ostheimer, Hofstra University

Abstract:
In 1954 Howson proved that in a free group, the intersection of two finitely generated subgroups is itself finitely generated. In this talk we will explore the extent to which Howson’s Theorem for free groups holds in finitely generated free metabelian groups. We show that in the latter context the intersection of finitely generated subgroups may not itself be finitely generated. On the other hand, when we define a suitable generalization of a finite generating set (which we call a “finite description”), we find that the intersection of two subgroups with finite descriptions has a finite description of its own, and that this finite description is computable. In this way we establish the decidability of the problem of determining whether the intersection of two finitely generated subgroups of a finitely generated free metabelian group is itself finitely generated as well as that of determining whether the intersection is trivial. Our decidability results actually hold in the more general context of the wreath product of two finitely generated free abelian groups.

This is joint work with Gilbert Baumslag and Chuck Miller.