Groups acting on trees with Tits’ independence property (P): with an appendix by Stephan Tornier

Local actions (actions of a vertex stabiliser on the neighbours of that vertex) have become an important approach to group actions on trees since J. Tits’ introduction in 1970 of the independence property (P) and especially since a 2000 paper by M. Burger and Sh. Mozes. This ‘local-to-global’ approach has been critical in the development of the theory of totally disconnected locally compact groups because it allows the construction of additional group actions on trees while keeping control over the action of a vertex stabiliser, in a way that is not practical under the classical Bass–Serre approach. The majority of constructions of nonlinear nondiscrete locally compact simple groups use (P) and its generalisations. In this article we give a full classification and description of all closed group actions on trees with Tits’ independence property (P) using a new coherent theory for local actions that applies to all actions on trees. This theory is a ‘local action’ complement to classical Bass–Serre theory. On the one hand, our theory gives a decomposition of a group acting on a tree into a ‘local action diagram’ (a decorated graph that encodes all ‘local’ information), and on the other hand a construction of a group acting on a tree from a given local action diagram. One can read directly from the local action diagram whether the resulting group has certain properties, like geometric density, compact generation and simplicity.