Acylindrically hyperbolic groups

Abstract

We say that a group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class $C_{geom}$ introduced by Hamenstadt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acylindrically hyperbolic groups for future use.

Lemma 6.8 is added

Cited by in corpus (5)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
• On topologizable and non-topologizable groups
• Partial periodic quotient of groups acting on a hyperbolic space
• Transitivity degrees of countable groups and acylindrical hyperbolicity