Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups

Abstract

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple $(\mathit{G},\mathit{X},\mathit{H})$ consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H that is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of $\mathrm{Out}({\mathit{F}}_{\mathit{n}})$, and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich–Rafi and Dowdall–Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.