Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G, X, 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 Out (F 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.