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.
Keywords
tree of words; rooted tree; automorphism of a rooted tree; group of automorphisms of a rooted tree
References
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Instytut Matematyki, Politechnika Śląska Poland
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