Simon M. Smith
March 6, 2012
Abstract. If is a group of permutations of a set , then the suborbits of are the orbits of point-stabilizers acting on . The cardinalities of these suborbits are the subdegrees of . Every infinite primitive permutation group with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph with vertex set , and there is consequently a natural action of on the ends of . We show that if is closed in the permutation topology of pointwise convergence, then the structure of is determined by the length of any orbit of acting on the ends of . 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 are the rough ends of , and so our result is equivalent to saying that the structure of a closed primitive group whose subdegrees are all finite is determined by the length of any orbit of on its rough ends.