On convex hulls and the quasiconvex subgroups of Fm×ℤn

Abstract In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of Fm×ℤn on Tree ×ℝ n ${\mathrm {Tree}\times \mathbb {R}^n}$ , every quasiconvex subgroup of Fm×ℤn is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of Fm×ℤn are precisely those of the form A×B, where A≤F m ${A\le F_m}$ is finitely generated, and B≤ℤ n ${B\le \mathbb {Z}^n}$ .