Abelian state-closed subgroups of automorphisms of $m$-ary trees
Andrew M. Brunner, Said N. Sidki · Groups Geometry and Dynamics · 2010
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^* 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. Furthermore, the group A splits over its torsion subgroup. We study in detail the case where A^* is additively a cyclic ℤ_m [[ 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.