Simon M. Smith (postdoc '12)
Journal: Journal of Group Theory
Abstract: If G is a group of permutations of a set , then the suborbits of G are the orbits of point-stabilizers G acting on . 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 with vertex set , and there is consequently a natural action of G on the ends of . 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 . Examining the ends of a Cayley graph of a finitely-generated group to determine the structure of the group is often fruitful. B. Krn and R. G. Mller 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 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.
URL: http://dx.doi.org/10.1515/jgt.2011.108