New 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 called locally-L if the permutation group induced by the action of the vertex-stabiliser G_v on the neighbourhood of a vertex v in Γ is permutation isomorphic to L. The maximum growth of | G_v | as a function of |V Γ | for locally-L pairs $( Γ,G)$ is called the graph growth of L. Recently, we have proven that if a transitive permutation group on a finite set Ω admits a proper block B such that the pointwise stabiliser of ΩB in L is non-trivial, then the graph growth of L is exponential. In this talk, we discuss core ideas behind this result and illustrate its impact by providing an overview of graph growth types for transitive permutation groups of low degree.

This is joint work with Gabriel Verret.