THE GROUP OF BALANCED AUTOMORPHISMS OF A SPHERICALLY HOMOGENEOUS ROOTED TREE

Abstract. Let X ∗ be a tree of words over the changing alphabet (X0, X1,...) with Xi = {0, 1,...,mi − 1}, mi> 1. We consider the group Aut(X∗) of automorphisms of a tree X∗. A cyclic automorphism of X ∗ is called constant if its root permutations at any two words from the same level of X ∗ coincide. In this paper we introduce the notion of a balanced automorphism which is obtained from a constant automorphism by changing root permutations at all words ending with an odd letter for their inverses. We show that the set of all balanced automorphisms forms a subgroup of Aut(X∗) if and only if 2- mi implies mi+1 = 2 for i = 0, 1,.... We study, depending on a branch index of a tree, the algebraic properties of this subgroup. 1