Geodesic
Abelian state-closed subgroups of automorphisms of $m$-ary trees
Andrew M. Brunner1 ; Said N. Sidki2
1University of Wisconsin-Parkside, Kenosha, USA
2Universidade de Brasília, Brazil
Groups, geometry, and dynamics, Tome 4 (2010) no. 3, pp. 455-472

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DOI

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.
DOI : 10.4171/ggd/91
Classification : 20-XX, 00-XX
Mots-clés : Automorphisms of trees, state-closed groups, self-similar groups, abelian groups, topological closure, <var>p</var>-adic integers, pro-<var>p</var> groups

Andrew M. Brunner  1   ; Said N. Sidki  2

1 University of Wisconsin-Parkside, Kenosha, USA
2 Universidade de Brasília, Brazil 
Andrew M. Brunner; Said N. Sidki. Abelian state-closed subgroups of automorphisms of $m$-ary trees. Groups, geometry, and dynamics, Tome 4 (2010) no. 3, pp. 455-472. doi: 10.4171/ggd/91
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     year = {2010},
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