Orbital graphs of infinite primitive permutation groups

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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 stabiliser Gα 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 connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in a previous paper by the author.