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

TL;DR

This paper characterizes abelian, self-similar subgroups of automorphisms of m-ary trees, showing their closure forms a finitely presented module over m-adic integers and analyzing their structure in special cases.

Contribution

It proves that the closure of such subgroups is a finitely presented module over Z_m[[x]] and classifies their structure when torsion-free or cyclic, especially for prime m.

Findings

Closure is a finitely presented Z_m[[x]]-module.

Torsion-free closure is a finitely generated pro-m group.

In prime m cases, groups are conjugate to specific types.

Abstract

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.