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

Abstract

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

Undistorted purely pseudo-Anosov groups

“…We note that the general strategy of our proof shares some features with the proof of Theorem 1.1 of [KMT17] regarding an analogous class of subgroups of right-angled Artin groups, though the techniques are quite different.…”

On strongly quasiconvex subgroups

“…Theorem 8.6 (Theorem 1.1, Theorem 5.2, and Corollary 6.2 in [KMT17]). Let Γ be a simplicial, finite, connected graph such that Γ does not decompose as a nontrivial join.…”

“…Moreover, stable subgroups of mapping class groups are precisely the convex cocompact subgroups defined by Farb-Mosher in [FM02] and the such subgroups of mapping class groups is a primary motivation for the concept of stable subgroups of arbitrary finitely generated groups (see [DT15b]). Stable subgroups have many similar properties to quasiconvex subgroups of hyperbolic groups and the study of stable subgroups has received much attention in recent years (see [KMT17], [ADT17], [AMST], [ABD], [CH17]).…”

Boundary Convex Cocompactness and Stability of Subgroups of Finitely Generated Groups

“…(3) For the right-angled Artin group A(Γ) of a finite simplicial graph Γ which is not a join, Koberda-Mangahas-Taylor [KMT14] proved that stability for H < A(Γ) is equivalent to H being finitely generated and purely loxodromic, i.e. each element acts loxodromically on the associated extension graph Γ e , a curve graph analogue.…”

“…In [ADT16], the authors prove this stability property pulls back to genuine stability in Out(F). We summarize these results in the following theorem: KMT14,ADT16]). Suppose that the pair H < G satisfies one of the following conditions:…”