Self-similar abelian groups and their centralizers

[摘要] We extend results on transitive self-similar abelian subgroups of the group of automorphisms Am\mathcal{A}_mAm​ of an mmm-ary tree Tm\mathcal{T}_mTm​ by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has s≥1s\geq 1s≥1 orbits. We prove that such a group AAA embeds in a self-similar abelian group A∗A^*A∗ which is also a maximal abelian subgroup of Am\mathcal{A}_mAm​. The construction of A∗A^{*}A∗ is based on the definition of a free monoid Δ\DeltaΔ of rank sss of partial diagonal monomorphisms of Am\mathcal{A}_mAm​. Precisely, A∗=Δ(B(A))‾A^{*} = \overline{\Delta(B(A))}A∗=Δ(B(A))​, where B(A)B(A)B(A) denotes the product of the projections of AAA in its action on the different sss orbits of maximal subtrees of Tm\mathcal{T}_mTm​, and bar denotes the topological closure. Furthermore, we prove that if AAA is non-trivial, then A∗=CAm(Δ(A))A^{*} = C_{\mathcal{A}_m} (\Delta(A))A∗=CAm​​(Δ(A)), the centralizer of Δ(A)\Delta(A)Δ(A) in Am\mathcal{A}_mAm​. When AAA 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 Δ\DeltaΔ-invariant for s=2s=2s=2. In the final section, we introduce for m=ns≥2m=ns \geq 2m=ns≥2, a generalized adding machine aaa, an automorphism of Tm\mathcal{T}_{m}Tm​, and show that its centralizer in Am\mathcal{A}_{m}Am​ to be a split extension of ⟨a⟩∗\langle a \rangle^{*}⟨a⟩∗ by As\mathcal{A}_sAs​. We also describe important Zn[As]\mathbb{Z}_n [\mathcal{A}_s]Zn​[As​] submodules of ⟨a⟩∗\langle a\rangle^{*}⟨a⟩∗.

[学科分类] 神经科学

[关键词] Groups acting on rootedm-tree;self-similar abelian groups;centralizers of abelian groups