Acylindrically hyperbolic group

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. 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").
Formal definition.
Acylindrical action.
Let "G" be a group with an isometric action on some geodesic hyperbolic metric space "X". This action is called acylindrical if for every formula_1 there exist formula_2 such that for every formula_3 with formula_4 one has
If the above property holds for a specific formula_1, the action of "G" on "X" is called "R"-acylindrical. 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 formula_7 such that their fixed point sets formula_8 and formula_9 are disjoint. 
It is known (Theorem 1.1 in ) 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".
Acylindrically hyperbolic group.
A group "G" is called acylindrically hyperbolic if "G" admits a non-elementary acylindrical isometric action on some geodesic hyperbolic metric space "X".
Equivalent characterizations.
It is known (Theorem 1.2 in ) that for a group "G" the following conditions are equivalent: