William Cohen

3-manifold groups acting acylindrically on trees

It was shown by Balasubramanya that any acylindrically hyperbolic group must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concept of an acylindrically arboreal group and consider some examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees. I will then present a characterisation of when exactly the fundamental groups of closed and orientable hyperbolic 3-manifolds act acylindrically and non-elementarily on trees based entirely on the geometry of the 3-manifolds in question.