The group of automorphisms of a one-rooted -ary tree admits a diagonal monomorphism which we denote by . Let be an abelian state-closed (or self-similar) subgroup of . We prove that the combined diagonal and tree-topological closure of is additively a finitely presented []-module, where is the ring of -adic integers. Moreover, if is torsion-free then it is a finitely generated pro- group. Furthermore, the group splits over its torsion subgroup. We study in detail the case where is additively a cyclic []-module, and we show that when m is a prime number then 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 -ary trees. Groups Geom. Dyn. 4 (2010), no. 3, pp. 455–472DOI 10.4171/GGD/91