Groups acting on trees with prescribed local actions

Abstract:

Actions on trees are ubiquitous in group theory.  The standard approach to describing them is known as Bass–Serre theory, which presents the group acting on the tree as assembled from its vertex and edge stabilizers.  However, a different approach emerges if instead of considering vertex and edge stabilizers as a whole, we focus on local actions, that is, the action of a vertex stabilizer only on the immediate neighbours of that vertex.  Groups acting on trees defined by their local actions are especially important as a source of examples of simple totally disconnected locally compact groups, with a history going back to a 1970 paper of Tits.  I will go through some highlights of this theory and then present some recent joint work with Simon Smith: we develop a counterpart to Bass–Serre theory for local actions, which describes all possible local action structures of group actions on trees. The talk is partly based on the following preprint of Simon Smith and the speaker: https://arxiv.org/abs/2002.11766.