Self-similar abelian groups and their centralizers

Abstract

We extend results on transitive self-similar abelian subgroups of the group of automorphisms \mathcal{A}_m of an m -ary tree \mathcal{T}_m by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has s\geq 1 orbits. We prove that such a group A embeds in a self-similar abelian group A^* which is also a maximal abelian subgroup of \mathcal{A}_m . The construction of A^{*} is based on the definition of a free monoid \Delta of rank s of partial diagonal monomorphisms of \mathcal{A}_m . Precisely, A^{*} = \overline{\Delta(B(A))} , where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of \mathcal{T}_m , and bar denotes the topological closure. Furthermore, we prove that if A is non-trivial, then A^{*} = C_{\mathcal{A}_m} (\Delta(A)) , the centralizer of \Delta(A) in \mathcal{A}_m . When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also \Delta -invariant for s=2 . In the final section, we introduce for m=ns \geq 2 , a generalized adding machine a , an automorphism of \mathcal{T}_{m} , and show that its centralizer in \mathcal{A}_{m} to be a split extension of \langle a \rangle^{*} by \mathcal{A}_s . We also describe important \mathbb{Z}_n [\mathcal{A}_s] submodules of \langle a\rangle^{*} .