Groups acting on trees with Tits' independence property (P)

Abstract: A 1970 article of J. Tits concerning groups acting on trees introduced an independence property $(\mathrm{P})$ as a condition to produce the first examples of nonlinear nondiscrete locally compact simple groups, answering a question of J. P. Serre. This property has become very important in the recent development of the theory of totally disconnected, locally compact (t.d.l.c.) groups, with the majority of new constructions of compactly generated simple t.d.l.c. groups using $(\mathrm{P})$ or related ideas. In this paper we aim to advance the local-to-global theory of groups acting on trees by developing a `local action' complement to classical Bass--Serre theory. We describe, for a closed group $G$ of automorphisms of a (not necessarily locally finite) tree $T$ something called a local action diagram: a graph decorated with the local actions of $G$. A local action diagram plays a role in our theory that is analogous to a graph of groups in Bass--Serre theory. In place of the universal cover of a graph of groups, we define the universal group of a local action diagram. In this context, the groups $\mathbf{U}(F)$ and $\mathbf{U}(F_1, F_2)$ play analogous roles to the HNN extension and amalgamated free product respectively in Bass--Serre theory. We then show how to determine whether the universal group has certain properties, such as geometric density, compact generation and simplicity, directly from the local action diagram. Our theory allows us to completely describe all closed groups of automorphisms of trees with Tits' independence property $(\mathrm{P})$: they are precisely the universal groups of local action diagrams.