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

Abstract

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.