Orbital digraphs of infinite primitive permutation groups

If G is a group acting on a set Ω, and α, β ∈ Ω, the digraph whose vertex set is Ω and whose arc set is the orbit (α, β)G is called an orbital digraph of G. Each orbit of the stabilizer G α acting on Ω is called a suborbit of G. A digraph is locally finite if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph Γ has more than one end if there exists a finite set of vertices X such that the induced digraph Γ\X contains at least two infinite connected components; if there exists such a set containing precisely one element, then Γ has connectivity one. In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivityone orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterized in a previous paper by the author

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

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.Comment: Survey article based on Simon M Smith's lecture at Groups St Andrews 202