Strong quasiconvexity, stability, and lower relative divergence in right-angled Artin groups

TL;DR

This paper establishes the equivalence of strong quasiconvexity and stability in right-angled Artin groups and characterizes certain subgroups using quadratic lower relative divergence, advancing subgroup classification.

Contribution

It proves the equivalence of strong quasiconvexity and stability in right-angled Artin groups and characterizes non-trivial stable subgroups via quadratic divergence.

Findings

Strong quasiconvexity and stability are equivalent in right-angled Artin groups.

Non-trivial stable subgroups are characterized by quadratic lower divergence.

Results extend previous work on loxodromic subgroup characterization.

Abstract

Let $\Gamma$ be a simplicial, finite, connected graph such that $\Gamma$ does not decompose as a nontrivial join. We prove that two notions of strong quasiconvexity and stability are equivalent in the right-angled Artin group $A_\Gamma$ (except for the case of finite index subgroups). We also characterize non-trivial strongly quasiconvex subgroups of infinite index in $A_\Gamma$ (i.e. non-trivial stable subgroups in $A_\Gamma$) by quadratic lower relative divergence. These results strengthen the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.