Solutions to two open problems in geometric group theory

Abstract

We introduce a method for analyzing the convex hull of a set in non-positively curved piecewise Euclidean polygonal complexes and we apply this method to prove that, with the usual action of Fm x Zn on the metric product of the Cayley graph of Fm with Rn, every quasiconvex subgroup of Fm x Zn is convex. This answers the question whether a quasiconvex subgroup of a CAT(0) group is a CAT(0) group in the affirmative for the groups F m x Zn. We also prove bounded packing in a special class of polycyclic groups, and we introduce the notion of coset growth and provide a bound for the coset growth of uniform lattices in Sol.