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

TL;DR

This paper investigates bounded packing properties of subgroups in CAT(0) and Gromov hyperbolic spaces, proving positive results for certain cyclic subgroups and constructing counterexamples for others.

Contribution

It proves that cyclic subgroups generated by rank-1 isometries have bounded packing in CAT(0) groups and constructs finitely generated subgroups without bounded packing.

Findings

Cyclic subgroups generated by rank-1 isometries have bounded packing.

Existence of finitely generated subgroups of $ imes ext{F}_2$ without bounded packing.

Existence of finitely presented subgroups of CAT(0) groups without bounded packing.

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.