On transitive permutation groups with exponential graph growth

Let Γ be a finite connected graph and G a vertex-transitive group of its automorphisms. The pair (Γ, G) is said to be locally-L if the permutation group induced by the action of the vertex-stabiliser Gᵥ on the set of neighbours of a vertex v in Γ is permutation isomorphic to L. The maximum growth of |Gᵥ| as a function of |VΓ| for locally-L pairs (Γ, G) is called the graph growth of L. We prove that if L is a transitive permutation group on a set Ω admitting a nontrivial block B such that the pointwise stabiliser of Ω B in L is nontrivial, then the graph growth of L is exponential. This generalises several results in the literature on transitive permutation groups with exponential graph growth.