ABELIAN STATE-CLOSED SUBGROUPS OF AUTOMORPHISMS OF m-ARY TREES
نویسندگان
چکیده
The group Am of automophisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of Am. We prove that the recurrence and tree-topological closure A∗ of A is additively a finitely presented Zm [[x]]module where Zm is the ring of m-adic integers. Moreover, if A∗ is torsion-free then it is a finitely generated pro-m group. The group A splits over its torsion subgroup. We study in detail the case where A∗ corresponds to a cyclic Zm [[x]]-module and when m is a prime number, we show A∗ to be conjugate by a tree automorphism to one of two specific types of groups.
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