Groups acting on trees: constructions, computations and classifications

Abstract:

Following a motivation of groups acting on trees by situating them within the broader theory of all (locally compact) groups, I demonstrate that the ’global’ structure of such a group is innately connected to the ’local’ actions that vertex stabilisers induce on balls around the fixed vertex. This is illustrated by a particularly accessible class of groups acting on trees with ’prescribed’ local actions, following Burger-Mozes. Being defined solely in terms of finite permutation groups, these groups allow us to introduce computational methods to the world of locally compact groups: I will outline the capabilities of a recently developed GAP package that provides methods to create, analyse and find suitable local actions (joint work with Khalil Hannouch). Finally, I will outline a strategy to classify closed, vertex-transitive groups acting on trees.