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