Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups

TL;DR

This paper introduces the concept of A/QI triples in geometric group theory, analyzing their properties and examples, and establishing stability and boundary behavior results for these structures.

Contribution

It formalizes the notion of A/QI triples, explores their intersection and combination properties, and applies existing theorems to analyze their boundaries and stability.

Findings

H is stable under certain conditions

The Gromov boundary of the cone-off can be characterized

Examples include various convex cocompact subgroups

Abstract

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 which 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(Fn), 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.