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 F_m x Z^n on the metric product of a tree with R^n, every quasiconvex subgroup of F_m x Z^n 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 Z^n. 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.