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- W3203957495 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 $sgeq 1$ 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 $langle arangle^{}$." @default.
- W3203957495 created "2021-10-11" @default.
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- W3203957495 date "2023-01-27" @default.
- W3203957495 modified "2023-09-25" @default.
- W3203957495 title "Self-similar abelian groups and their centralizers" @default.
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- W3203957495 doi "https://doi.org/10.4171/ggd/710" @default.
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