An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams

TL;DR

This paper surveys the theory of local action diagrams, a new approach to understanding groups acting on trees that complements Bass-Serre Theory and describes groups with Tits' Independence Property.

Contribution

It introduces the theory of local action diagrams, providing a new framework to analyze groups acting on trees and describing all closed groups with Tits' Independence Property.

Findings

Developed a general theory of universal groups of local actions

Described all closed groups of automorphisms with Tits' Independence Property

Provided ideas for future research in the area

Abstract

The primary tool for analysing groups acting on trees is Bass--Serre Theory. It is comprised of two parts: a decomposition result, in which an action is decomposed via a graph of groups, and a construction result, in which graphs of groups are used to build examples of groups acting on trees. The usefulness of the latter for constructing new examples of `large' (e.g. nondiscrete) groups acting on trees is severely limited. There is a pressing need for new examples of such groups as they play an important role in the theory of locally compact groups. An alternative `local-to-global' approach to the study of groups acting on trees has recently emerged, inspired by a paper of Marc Burger and Shahar Mozes, based on groups that are `universal' with respect to some specified `local' action. In recent work, the authors of this survey article have developed a general theory of universal groups of local actions, that behaves, in many respects, like Bass--Serre Theory. We call this the theory of local action diagrams. The theory is powerful enough to completely describe all closed groups of automorphisms of trees that enjoy Tits' Independence Property (P). This article is an introductory survey of the local-to-global behaviour of groups acting on trees and the theory of local action diagrams. The article contains many ideas for future research projects.