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

Abstract

The group ${\mathcal{A}}_m$ 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 ${\mathcal{A}}_m$. We prove that the combined diagonal and tree-topological closure $A^{\ast }$ of $A$ is additively a finitely presented $Z_m$[[$x$]]-module, where $Z_m$ is the ring of $m$-adic integers. Moreover, if $A^{\ast }$ 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 $A^{\ast }$ is additively a cyclic $Z_m$[[$x$]]-module, and we show that when m is a prime number then $A^{\ast }$ is conjugate by a tree automorphism to one of two specific types of groups.