Subgroups of free metabelian groups

Abstract

In 1954 A. G. Howson proved that the intersection of two finitely generated subgroups of a free group is again finitely generated. Now the free metabelian subgroups of a free metabelian group of finite rank n are quite restricted. Indeed they are again of finite rank at most n. This suggests that there may be an analog of Howson’s theorem for free metabelian groups. This turns out not to be the case. The object of this paper is to explore such intersections in free metabelian groups and, more generally, in the wreath product of two free abelian groups. In such a wreath product we show, for instance, that there are algorithms to decide whether or not the intersection of two finitely subgroups is finitely generated or trivial. This leaves open the existence of algorithms to decide the same questions for finitely generated subgroups of finitely generated metabelian groups as a whole.