Loosely speaking, an action of a group on a tree is acylindrical if long enough paths must have small stabilisers. Groups admitting such actions form a natural subclass of acylindrically hyperbolic groups, and interesting an feature of acylindrical actions on trees is that many interesting properties are inherited from their vertex stabilisers. In order to make use of this, it is important to have some degree of control over these stabilisers. For example, can we ask for these stabilisers to be finitely generated, or even malnormal (or finite-height)? Even stronger, if our group is hyperbolic, can we ask for the stabilisers to be quasiconvex?

 

In this talk, I will introduce acylindrical actions and some stronger and related concepts, and discuss a method known as the Dunwoody—Sageev resolution that we can use to move between these concepts and provide positive answers to the above questions in some cases.