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We abstract the notion of an A/QI triple from a number of examples ingeometric group theory. Such a triple (G,X,H) consists of a group G acting on aGromov hyperbolic space X, acylindrically along a finitely generated subgroup Hwhich is quasi-isometrically embedded by the action. Examples include stronglyquasi-convex subgroups of relatively hyperbolic groups, convex cocompactsubgroups of mapping class groups, many known convex cocompact subgroups ofOut(Fn), and groups generated by powers of independent loxodromic WPD elementsof a group acting on a Gromov hyperbolic space. We initiate the study ofintersection and combination properties of A/QI triples. Under the additionalhypothesis that G is finitely generated, we use a method of Sisto to show thatH is stable. We apply theorems of Kapovich--Rafi and Dowdall--Taylor to analyzethe Gromov boundary of an associated cone-off. We close with some examples andquestions.

Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups