Not all acylindrically hyperbolic groups have universal acylindrical actions

Carolyn Abbott (Wisconsin)

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(Fn) for n >1. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated acylindrically hyperbolic group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. In this talk, I will answer this question in the negative, using Dunwoody’s example of an inaccessible group as a counterexample.