Orbital graphs of infinite primitive permutation groups

Abstract

If $G$ is a group acting on a set $\Omega$ and $\alpha, \beta \in \Omega$, the digraph whose vertex set is $\Omega$ and whose arc set is the orbit $(\alpha, \beta)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser $G_\alpha$ acting on $\Omega$ is called a {\it suborbit} of $G$. A digraph is {\em locally finite} if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph $\Gamma$ has more than one end if there exists a finite set of vertices $X$ such that the induced digraph $\Gamma \setminus X$ contains at least two infinite connected components; if there exists such a set containing precisely one element, then $\Gamma$ has {\em 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.