An application of the Local C(G,T) Theorem to a conjecture of Weiss

Let Γ be a connected G-vertex-transitive graph, let v be a vertex of Γ and let GvΓ(v) be the permutation group induced by the action of the vertex-stabilizer Gv on the neighbourhood Γ (v). The graph Γ is said to be G-locally primitive if GvΓ(v) is primitive. Weiss conjectured in 1978 that there exists a function f: N → N such that, if Γ is a connected G-vertex-transitive locally primitive graph of valency d and v is a vertex of Γ with /Gv/ finite, then /Gv/≤ f(d). As an application of the Local C(G,T) Theorem, we prove this conjecture when GvΓ(v) contains an abelian regular subgroup. In fact, we show that the point-wise stabilizer in G of a ball of Γ of radius 4 is the identity subgroup.