Existence and non-existence of bounded packing in CAT(0) spaces and Gromov hyperbolic spaces

Abstract: The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Hamenstadt. The second proof follows directly from some results of Dahmani-Guirardel-Osin and Sisto. Then using Mihailova's construction, we show the existence of a finitely generated subgroup of the direct product of two free groups $\mathbb F_2\times \mathbb F_2$ without the bounded packing property answering a question of Hruska-Wise. We also prove the existence of finitely presented subgroups of CAT(0) groups without bounded packing using Wise's {\em modified Rip's construction} and the {\bf 1-2-3} theorem of Baumslag, Bridson, Miller and Short.