Topology Seminar (Main Talk): Universal acylindrical actions

Seminar | August 30 | 4-5 p.m. | 3 Evans Hall

 Carolyn Abbott, UC Berkeley

 Department of Mathematics

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 \(\operatorname {Out}(\mathbb F_n)\) 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. Given a finitely generated group, one can look for an acylindrical action on a hyperbolic space in which all generalized loxodromic elements act loxodromically; such an action is called a universal acylindrical action. I will discuss recent results in the search for universal acylindrical actions, describing a class of groups for which it is always possible to construct such an action as well as an example of a group for which no such action exists.