Rough ends of infinite primitive groups

Abstract

If $G$ is a group of permutations of a set $Ω$, then the suborbits of $G$ are the orbits of point-stabilisers $G_α$ 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. Kr{ö}n and R. G. M{ö}ller have recently generalised the Cayley graph to what they call a {\it rough Cayley graph}, and they call the ends of this graph the {\it 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.