Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

François Dahmani (1) , Vincent Guirardel (2) , Denis Osin (3)

1 IF - Institut Fourier
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Department of Mathematics, Vanderbilt University

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of peripheral structures of relative hyperbolicity groups, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on CAT(0) spaces, fundamental groups of graphs of groups, etc. Although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

François Dahmani, Vincent Guirardel, Denis Osin. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Memoirs of the American Mathematical Society, 2017, 245 (1156), ⟨10.1090/memo/1156⟩. ⟨hal-00704880⟩