Rough ends of infinite primitive groups

TL;DR

This paper studies the structure of infinite primitive permutation groups with finite subdegrees, showing it is determined by the action on the ends of associated graphs, specifically the rough ends of the group.

Contribution

It establishes that the structure of closed primitive groups with finite subdegrees is determined by the orbit length on the rough ends of their associated graphs.

Findings

The structure of such groups is determined by the length of an orbit on the rough ends.

Ends of the graph correspond to rough ends of the group.

Group structure is linked to its action on rough ends.

Abstract

If $G$ is a group of permutations of a set $\Omega$, then the suborbits of $G$ are the orbits of point-stabilisers $G_\alpha$ acting on $\Omega$. 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 $\Gamma$ with vertex set $\Omega$, and there is consequently a natural action of $G$ on the ends of $\Gamma$. 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 $\Gamma$. Examining the ends of a Cayley graph of a finitely generated group to determine the structure of the group is often fruitful. B. Kr{\"o}n and R. G. M{\"o}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 $\Gamma$ 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.