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

TL;DR

This paper characterizes purely loxodromic subgroups of right-angled Artin groups, showing they are quasiconvex and exploring conditions for subgroups to be free and undistorted, with implications for mapping class groups.

Contribution

It establishes equivalent conditions for purely loxodromic subgroups to be quasiconvex and identifies milder conditions for subgroups to be free and undistorted in right-angled Artin groups.

Findings

Purely loxodromic subgroups are quasiconvex in right-angled Artin groups.

A milder condition ensures subgroups are free, undistorted, and finitely generated.

Results connect convex cocompactness in mapping class groups with subgroup embeddings in RAAGs.

Abstract

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 $\text{Mod}(S)$. 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 $\text{Mod}(S)$ and the way in which certain groups can embed in right-angled Artin groups.