In the mathematical subject of geometric group theory, an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space.
[1] This notion generalizes the notions of a hyperbolic group and of a relatively hyperbolic group and includes a significantly wider class of examples, such as mapping class groups and Out(Fn).
Let G be a group with an isometric action on some geodesic hyperbolic metric space X.
This action is called acylindrical[1] if for every
The notion of acylindricity provides a suitable substitute for being a proper action in the more general context where non-proper actions are allowed.
An acylindrical isometric action of a group G on a geodesic hyperbolic metric space X is non-elementary if G admits two independent hyperbolic isometries of X, that is, two loxodromic elements
It is known (Theorem 1.1 in [1]) that an acylindrical action of a group G on a geodesic hyperbolic metric space X is non-elementary if and only if this action has unbounded orbits in X and the group G is not a finite extension of a cyclic group generated by loxodromic isometry of X.
A group G is called acylindrically hyperbolic if G admits a non-elementary acylindrical isometric action on some geodesic hyperbolic metric space X.
It is known (Theorem 1.2 in [1]) that for a group G the following conditions are equivalent: