On locally semiprimitive graphs and a theorem of Weiss

Abstract

In this paper we investigate graphs that admit a group acting arc-transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup. This type of semiprimitive group is a generalisation of an affine group. We show that if the graph has valency coprime to six, then there is a bound on the order of the vertex stabilisers depending on the valency alone. We also prove a detailed structure theorem for the vertex stabilisers in the remaining case. This is a contribution to an ongoing project to investigate the validity of the Poto\v{c}nik-Spiga-Verret Conjecture.