Abelian state-closed subgroups of automorphisms of m-ary trees

Details Abelian state-closed subgroups of automorphisms of m-ary trees Abelian state-closed subgroups of automorphisms of m-ary trees

2009-06-18

arxiv.org/abs/0906.3450v2

Mathematics

Group Theory

14 pages

Scientific paper

The group A_{m} 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 A_{m}. We prove that the combined diagonal and tree-topological closure A* of A is additively a finitely presented Z_m [[x]]-module where Z_m 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 Z_m[[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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