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- Abstract. We consider the structure of groups that act on apn-regular tree in a vertex transitive way with the local action (ie the action of the vertex stabilizer on the link) isomorphic to the group of af¢ne transformations on a ¢nite af¢ne line … Mathematics Subject Classi¢cations (2000). Primary 20E; Secondary 05C25, 22E40, 20F32 … Let T be the q-regular tree, and H a closed subgroup of automorphisms of T acting transitively on the vertices T. The stabilizer of a vertex acts on the link of the vertex via a ¢nite permutation group say G < SymšqŽ which is independent of the given vertex. We say that the local action of H is given by the ¢nite permutation group G. Given a ¢nite permutation group G, Burger and Mozes ([BMa]) de¢ne a universal group UšGŽ < AutšTŽ containing a conjugate of every vertex transitive group H whose local action is given by G. Of special interest are the groups UšGŽ where G is a two-transitive …