Rylee Lyman - Bass-Serre theory: groups acting on trees (Lecture 2)

Graphs can be cut up into forests along vertices. Surfaces can be cut up into disks along simple closed curves. Haken manifolds can be cut up into 3-balls along 2-sided, embedded incompressible surfaces. Bass-Serre theory provides a common framework for reasoning about cutting up a group along subgroups. As such it is perhaps the main inductive tool in geometric group theory. In this mini-course, we will meet the main features of the theory: graphs of groups, their fundamental groups, their universal covering trees, and their reformulation (due to Scott and Wall) as graphs of spaces. We will also study covering spaces and morphisms of graphs of groups, less well-known but no less useful. At the course we will meet a handful of applications of Bass-Serre theory. Possible examples include JSJ decompositions of hyperbolic groups, the "Nielsen realization problem" for finite groups of outer automorphisms of free groups, and the Bestvina-Feighn combination theorem.