Automorphism groups of trees acting locally with affine permutations (Q1597464)

Let \(T\) be a \(q\)-regular tree, \(H\) a closed subgroup of automorphisms of \(T\) acting transitively on the vertices of \(T\). The stabilizer of a vertex acts on the link of the vertex via a finite permutation group \(G\leq\text{Sym}(q)\) which is independent of the given vertex. This action is said to be the local action of \(H\) given by the finite permutation group \(G\). Given a finite permutation group \(G\), \textit{M. Burger, S. Mozes} [Publ. Math., Inst. Hautes Étud. Sci. 92, 113-150 (2000; Zbl 1007.22012)] defined a universal group \(U(G)\leq\Aut(T)\) of every vertex transitive group \(H\) whose local action is given by \(G\). If \(G\) is a two-transitive permutation group, then \(G_0\) denotes the stabilizer of a point. Burger and Mozes considered the case where \(G_0\) is a non-Abelian simple group. In this paper the authors investigate the situation where \(G_0\) is very far from being non-Abelian simple. More precisely, they consider the group \(A\) of affine transformations acting on the affine line of some finite field. In this case \(A\) is a two-transitive permutation group and \(A_0\) is Abelian.