Hyperbolic Structures on groups

Abstract

It is customary in geometric group theory to study groups as metric spaces. The standard way to convert a group G into a geometric object is to fix a generating set X and endow the Cayley graph Ga(G,X) with the corresponding word metric. In joint work with Carolyn Abbott and Denis Osin, we introduced the set of hyperbolic structures on G, denoted H(G), which consists of equivalence classes of generating sets of G such that the corresponding Cayley graph is hyperbolic; these are ordered in a natural way according to the amount of information they provide about the group. Of special interest is the subset AH(G) of H(G) of acylindrically hyperbolic structures on G, i.e. hyperbolic structures corresponding to acylindrical actions.The question of accessibility of these posets is studied, and several classes of acylindrically hyperbolic groups are proved to be AH-accessible.

By utilizing the notions of hyperbolically embedded subgroups and projection complexes, I then prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph Ga(G,X) is a (non-elementary) quasi-tree and the action of G on Ga(G,X) is acylindrical. As an application, new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups are obtained.

Lastly, a particular question associated to quasi-parabolic hyperbolic structures is answered . Specifically, many examples of groups with finitely many quasi-parabolic structures are given.