Rough ends of infinite primitive groups

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If G is a group of permutations of a set Ω, then the suborbits of G are the orbits of point-stabilisers Gα acting on Ω. The cardinalities of these suborbits are the subdegrees of G. Every infinite primitive permutation group G with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph Γ with vertex set Ω, and there is consequently a natural action of G on the ends of Γ. We show that if G is closed in the permutation topology of pointwise convergence, then the structure of G is determined by the length of any orbit of G acting on the ends of Γ. Examining the ends of a Cayley graph of a finitely generated group to determine the structure of the group is often fruitful. B. Krön and R. G. Möller have recently generalised the Cayley graph to what they call a rough Cayley graph, and they call the ends of this graph the rough ends of the group. It transpires that the ends of Γ are the rough ends of G, and so our result is equivalent to saying that the structure of a closed primitive group G whose subdegrees are all finite is determined by the length of any orbit of G on its rough ends.