The geometry of purely loxodromic subgroups of right-angled Artin groups

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(\Gamma )$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\mathrm {Mod}(S)$. In particular, such subgroups are quasiconvex in $A(\Gamma )$. In addition, we identify a milder condition for a finitely generated subgroup of $A(\Gamma )$ that guarantees it is free, undistorted, and retains finite generation when intersected with $A(\Lambda )$ for subgraphs $\Lambda$ of $\Gamma$. These results have applications to both the study of convex cocompactness in $\mathrm {Mod}(S)$ and the way in which certain groups can embed in right-angled Artin groups.