Abstract
We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple 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 , 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.
Citation
Carolyn R. Abbott. Jason F. Manning. "Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups." Michigan Math. J. 74 (2) 357 - 402, April 2024. https://doi.org/10.1307/mmj/20216112
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