Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a G-vertex-transitive graph Γ. In the main result the group G is quasiprimitive or biquasiprimitive on the vertices of Γ, and we obtain a genuine reduction to the case where G is a non-abelian simple group.Using normal quotient techniques developed by the first author, the main theorem applies to general G-vertex-transitive graphs which are G-locally primitive (respectively, G-locally quasiprimitive), that is, the stabiliser Gα of a vertex α acts primitively (respectively quasiprimitively) on the set of vertices adjacent to α. We discuss how our results may be used to investigate conjectures by Richard Weiss (in 1978) and the first author (in 1998) that the order of Gα is bounded above by some function depending only on the valency of Γ, when Γ is G-locally primitive or G-locally quasiprimitive, respectively.