Trees: with a view towards Outer space
Trees are graphs without loops. Group actions on trees appear in geometric group theory and geometric topology through free group actions on trees, JSJ decompositions of 3-manifolds, and as the simplest examples of Bruhat-Tits buildings. More generally, \(\mathbb{R}\)-trees are metric spaces that look like trees but, unlike their simplicial cousins, need not have a discrete set of branch points. They appear as dual objects to codimension one foliations, and as rescaled limits of actions on hyperbolic spaces (e.g. in the boundary of Teichmüller space).
The aim of this course is to look at group actions on trees with a view towards understanding spaces of actions on trees and the algebraic approach to JSJ theory. This includes \(\mathbb{R}\)-trees, Bass-Serre theory, length functions, and gluing constructions (commonly called graphs of actions). We will then look at spaces of group actions on trees, such as Culler and Vogtmann's Outer space. This can be explored in the wider context of deformation spaces, and we will prove that such spaces are contractible. Depending on time and the interests of people that are attending, we can then go into more detail about JSJ decompositions or look at other specialist topics such as Rips' theorem, the Bestvina-Paulin construction, or Levitt's \(\mathbb{R}\)-tree description of BNS invariants.