On regular graphs,V

Let Γ₃ be an infinite regular tree of valence 3. There exist subgroups B of Aut (Γ₃) which are 5-regular on Γ₃, i.e., sharply transitive on the set of 5-arcs of Γ₃. We prove that any two such subgroups are conjugate in Aut (Γ₃). The pair (Γ₃, B) is a universal 5-regular action in the sense that if (G, A) is a pair consisting of a cubical graph G and a 5-regular subgroup A of automorphisms of G then (G, A) can be “covered” by (Γ₃, B) in a certain natural way.