Groups, Geometry, and Dynamics

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Volume 4, Issue 3, 2010, pp. 455–472
DOI: 10.4171/GGD/91

Published online: 2010-06-16

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

Andrew M. Brunner[1] and Said N. Sidki[2]

(1) University of Wisconsin-Parkside, Kenosha, USA
(2) Universidade de Brasília, Brazil

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.

Keywords: Automorphisms of trees, state-closed groups, self-similar groups, abelian groups, topological closure, p-adic integers, pro-p groups

Brunner Andrew, Sidki Said: Abelian state-closed subgroups of automorphisms of m-ary trees. Groups Geom. Dyn. 4 (2010), 455-472. doi: 10.4171/GGD/91