Georgia Institute of Technology College of Sciences

A group is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. This class encompasses many examples of interest: hyperbolic and relatively hyperbolic groups, Out(F_n) for n>1, all but finitely many mapping class groups, most fundamental groups of 3-manifolds, groups acting properly on proper CAT(0) spaces and containing rank 1 elements, 1-relator groups with at least 3 generators, etc. On the other hand, many results known for these particular classes can be naturally generalized in the context of acylindrically hyperbolic groups. In my talk I will survey some recent progress in this direction. The talk is partially based on my joint papers with F. Dahmani, V. Guirardel, M.Hull, and A. Minasyan.