arXiv subjects

Said N. Sidki

Publications and source records attributed to Said N. Sidki.

13 recordsLinked to original sources

arXiv2021

Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms $\mathcal{A}_m$ of an $m$-ary tree $\mathcal{T}_m$ in \cite{BS}, 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 $Δ$ of rank $s$ of partial diagonal monomorphisms of $\mathcal{A}_m$, which is used to determine the structure of $C_{\mathcal{A}_m}(A)$, the centralizer of $A$ in $\mathcal{A}_m$. Indeed, we prove $A^*=C_{\mathcal{A}_m} (Δ(A))=\overline{ Δ({B(A)})}$, 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. 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 $Δ$-invariant for $s=2$. Finally, we focus on self-similar cyclic groups of automorphisms of $\mathcal{T}_m$ and compute their centralizers when $m=4.$

math.GR↗

arXiv2020

Intransitive Self-similar Groups

A group is said to be self-similar provided it admits a faithful state-closed representation on some regular $m$-tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level of the tree. A standard approach for constructing a transitive self-similar representation of a group has been by way of a single virtual endomorphism of \ the group in question. Recently, it was shown that this approach when applied to the restricted wreath product $% \mathbb{Z}\wr \mathbb{Z}$ could not produce a faithful transitive self-similar representations for any $m\geq 2$ (see, \cite{DS}). In this work we study state-closed representations without assuming the transitivity condition. This general action is translated into a set of virtual endomorphisms corresponding to the different orbits of the action on the first level of the tree. In this manner, we produce faithful self-similar representations, some of which are also finite-state, for a number of groups such as $\mathbb{Z}^ω$, $\mathbb{Z}\wr \mathbb{Z}$ and $(\mathbb{Z} \wr \mathbb{Z}) \wr C_{2}$.

math.GR↗

arXiv2018

Self-similar products of groups

We address the problem of determining the class of self-similar groups, and in particular its closure under restricted direct products. We show that the group $\mathbb Z^{(ω)}$ is self-similar, that $G^{(ω)}\rtimes C_2$ is self-similar whenever $G$ is, and that permutational wreath products of a finite abelian group with a self-similar group are self-similar.

math.GR↗

arXiv2018

On Self-Similar Lie Algebras and Virtual Endomorphisms

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.

math.RA↗

arXiv2017

Self-similar groups of type FP_{n}

We construct new classes of self-similar groups : S-aritmetic groups, affine groups and metabelian groups. Most of the soluble ones are finitely presented and of type FP_{n} for appropriate n.

math.GR↗

arXiv2017

On self-similarity of wreath products of abelian groups

We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we construct a self-similar group $G=BwrC_{2}$ where $B$ is free abelian of infinite rank.

math.GR↗

arXiv2015

Affine Automorphisms of Rooted Trees

We introduce a class of automorphisms of rooted $d$-regular trees arising from affine actions on their boundaries viewed as infinite dimensional vector spaces. This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from this class and show that it is isomorphic to the index 2 extension of the rank 2 lamplighter group $\mathbb Z_2^2\wr\mathbb Z$.

math.GR↗

arXiv2015

On state-closed representations of restricted wreath product of groups of type G_{p,d}=C_{p}wrC^{d}

Let $G_{p,d}$ be the restricted wreath product $C_{p} wr C^{d}$ where $C_{p}$ is a cyclic group of order a prime $p$ and $C^{d}$ a free abelian group of finite rank $d$. We study the existence of faithful state-closed (fsc) representations of $G_{p,d}$ on the $1$-rooted $m$-ary tree for some finite $m$. The group $G_{2,1}$, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for $d \geq 2$ there are no fsc representations of $G_{p,d}$ on the $p$ -adic tree. We characterize all fsc representations of $G=G_{p,1}$ on the $p$-adic tree where the first level stabilizer of the image of $G$ contains its commutator subgroup. Furthermore, for $d \geq 2$, we construct uniformly fsc representations of $G_{p,d}$ on the $p^{2}$ -adic tree and exhibit concretely the representation of $G_{2,2}$ on the $4$-tree as a finite-state automaton group.

math.GR↗

arXiv2014

On the conjugacy problem for finite-state automorphisms of regular rooted trees

We study the conjugacy problem in the automorphism group $Aut(T)$ of a regular rooted tree $T$ and in its subgroup $FAut(T)$ of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in $Aut(T)$ if and only if they are conjugate in $FAut(T)$, and that this problem is decidable. We prove that both these conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable.

math.GR↗

arXiv2009

Functionally recursive rings of matrices-Two examples

We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log(2)/log(a) where a=1/2(1+sqrt(5)).

math.RA↗

arXiv2009

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

The group A_{m} of automophisms 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 A_{m}. 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. The group A splits over its torsion subgroup. We study in detail the case where A* corresponds to a cyclic Z_m[[x]]-module and when m is a prime number, we show A* to be conjugate by a tree automorphism to one of two specific types of groups.

math.GR↗

arXiv2009

On Commutativity and Finiteness in Groups

The second author introduced notions of weak permutability and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite groups. Two groups H,K weakly commute provided there exists a bijection f: H -> K which fixes the identity element and such that h commutes with its image h^f for all h in H. The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H,K.

math.GR↗

arXiv2006

The Automorphism Tower of Groups acting on Rooted Trees

The group of isometries W of a regular rooted tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in W. This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group H studied by N. Gupta and the second author. In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as G and H. We describe this tower for all groups acting on the binary rooted tree which decompose as infinitely iterated wreath products. Furthermore, we describe fully the towers of G and H. More precisely, the tower of G is infinite countable, and the terms of the tower are 2-groups. Quotients of successive terms are infinite elementary abelian 2-groups. In contrast, the tower of H has length 2, and its terms are {2,3}-groups. We show that the quotient aut^2(H)/aut(H) is an elementary abelian 3-group of countably infinite rank, while aut^3(H)=aut^2(H).

math.GR↗