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

spiga, pablo;

Let Γ be a connected G-vertex-transitive graph, let v be a vertex of Γ and let GΓ(v) 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 GΓ(v) 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 GΓ(v)
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.