Automorphisms of spherically transitive groups of automorphisms of a rooted tree (Q2740291)

Let \((T,x_0)\) be a spherically homogeneous tree (all its vertices which are equidistant from the root \(x_0\) of the tree \(T\) have the same valency). A subgroup \(G\subseteq\Aut(T)\) is called spherically transitive (ST-group) if \(G\) acts transitively on every layer of the tree (i.e. on each subset of vertices which are equidistant from the root of the tree). In the paper, ST-groups with non-trivial costabilizers of all vertices are studied. In particular, it is proved that under some restrictions the following equality holds: \(\Aut(G)=N_{\Aut(T)}(G)\). It is also shown that the class of groups under study is wide enough, for instance, it contains the Grigorchuk group.