The group Am of automorphisms 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 combined diagonal 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. Furthermore, the group A splits over its torsion subgroup. We study in detail the case where A* is additively a cyclic ℤm[[x]]-module, and we show that when m is a prime number then A* is conjugate by a tree automorphism to one of two specific types of groups.
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Andrew M. Brunner, Said N. Sidki, Abelian state-closed subgroups of automorphisms of <var>m</var>-ary trees. Groups Geom. Dyn. 4 (2010), no. 3, pp. 455–472DOI 10.4171/GGD/91