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

作者：Brunner Andrew M^*; Sidki Said N

来源：Groups, Geometry, and Dynamics, 2010, 4(3): 455-472.

DOI：10.4171/GGD/91

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

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 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.

• 出版日期2010