Restricted Wreath Product Research Articles

On reversible automata generating lamplighter groups

For any nontrivial abelian group X we construct a reversible automaton such that its set of states and alphabet are identified with X, transition and output functions are defined via the left regular action and its group splits into the restricted wreath product X≀Z, i.e. is a lamplighter group.

Haagerup property for wreath products constructed with Thompson's groups

Using recent techniques introduced by Jones, we prove that a large family of discrete groups and groupoids have the Haagerup property. In particular, we show that if \Gamma is a discrete group with the Haagerup property, then the permutational restricted wreath product \bigoplus_{\mathbb{Q}_2}\Gamma \rtimes V obtained from the group \Gamma and the usual action of Richard Thompson's group V on the dyadic rational \mathbb{Q}_2 of the unit interval has the Haagerup property.

Quantifying conjugacy separability in wreath products of groups

Abstract We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of A and B, the growth of $\mathcal{C}$-cyclic subgroup separability of B and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterization of when $A \wr B$ is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.

BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS

AbstractWe extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

A Higher Version of Zappa Products for Monoids

For arbitrary monoids $A$ and $B$, a presentation for the restricted wreath product of $A$ by $B$ that is known as the semi-direct product of $A^{\oplus B}$ by $B$ has been widely studied (see, for instance, \cite{Cevik,Howie-Ruskuc,Johnson,Meldrum}). After that a presentation for the Zappa product of $A$ by $B$ is defined (\cite{Brin,Lavers}) which can be thought as the mutual semidirect product of given these two monoids under a homomorphism $\psi : A \rightarrow \tau(B)$ and an anti-homomorphism $\delta : B \rightarrow \tau(A)$ into the full transformation monoid on $B$, resp. on $A$ (cf. \cite{Kunze}). As a next step of these above results, by considering the monoids $A^{\oplus B}$ and $B^{\oplus A}$, we will first introduce an extended version (generalization) of the Zappa product and then we will prove the existence of an implicit presentation for this new product. Furthermore we will present some other outcomes of the main theories in terms of finite and infinite cases, and also in terms of groups. At the final part of this paper we will point out some possible future problems related to this subject.

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 Z≀Z could not produce a faithful transitive self-similar representations for any m≥2 (see [5]). 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 Zω, Z≀Z and (Z≀Z)≀C2.

Twisted Burnside–Frobenius theorem and $R_infty$-property for lamplighter-type groups

We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $\varphi$ has infinite Reidemeister number $R(\varphi)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(\varphi)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[\rho]\mapsto[\rho\circ\varphi]}$.

Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band

Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not faithfully). Also commutator of such group and its minimal generating set. Also center of such products was investigated. Also fundamental group of orbits of a Morse function $f:M\to \mathbb{R}$ defined on a Mebius band $M$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$ is investigated by us. The paper describes precise algebraic structure of the group $\pi_1 O(f)$. A minimal set of generators for the group of orbits of functions ${{\pi }_{1}}({{O}_{f}},f)$ arising under the action of diffeomorphisms group stabilizing the function $f$ and stabilizing $\partial M$ is found. The the Morse function $f$ has critical sets with one saddle point. The quotient group of restricted wreath products by its commutator was found. The generic sets of commutator of wreath product were investigated. Minimal generating set for this group and for commutator of group are found. This paper after previous Arxiv versions from 2019 \cite{SkArM, SkArM3} with previous title "Minimal generating set and structure of wreath product of groups with non-faithful action, comutator subgroup of wreath product and the fundamental group of orbit of Morse function $\pi_1 O(f)$" was published \cite{SkRendi}.

Generating Sets and a Structure of the Wreath Product of Groups with Non-Faithful Group Action

Given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. We generalize the result presented in the book of J. Meldrum [11] also the results of A. Woryna [4]. The quotient group of the restricted and unrestricted wreath product by its commutator is found. The generic sets of commutator of wreath product were investigated. The structure of wreath product with non-faithful group action is investigated. We strengthen the results from the author [17, 19] and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. We generalise the results of Meldrum J. [11] about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group A does not have to act faithfully). The commutator of such a group, its minimal generating set and the center of such products has been investigated here. The minimal generating sets for new class of wreath-cyclic geometrical groups and for the commutator of the wreath product are found.

Permanence properties of the second nilpotent product of groups

We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath product constructed from it is unitarizable if and only if the acting group is amenable.

Some rigidity results for II1 factors arising from wreath products of property (T) groups

We show that any infinite collection (Γn)n∈N of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic infinite product rigidity phenomenon. If Λ is an arbitrary group such that L(⊕n∈NΓn)≅L(Λ) then there exists an infinite direct sum decomposition Λ=(⊕n∈NΛn)⊕A with A icc amenable or trivial such that, for all n∈N, up to amplifications, we have L(Γn)≅L(Λn) and L(⊕k≥nΓk)≅L((⊕k≥nΛk)⊕A). The result is sharp and complements the previous finite product rigidity property found in [16]. Using this we provide an uncountable family of restricted wreath products Γ≅Σ≀Δ of icc, property (T) groups Σ, Δ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras L(Γ). Along the way we highlight several applications of these results to the study of rigidity in the C⁎-algebra setting.

Reidemeister classes in lamplighter-type groups

We prove that for any automorphism of the restricted wreath product and the Reidemeister number is infinite (the property ). For and , where p > 3 is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property . For these groups and , where m is relatively prime to 6, we prove the twisted Burnside-Frobenius theorem (TBFTf): if , then it is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action .

New Properties over a New Type of Wreath Products on Monoids

In [1], a new consequence of the (restricted) wreath product for arbitrary monoids A and B with an underlying set . Let us denote it by . Actually, in the same reference, it has been also defined the generating and relator sets for , and then proved some finite and infinite cases about it. In this paper, by considering the product, we show Green’s relations L and R as well as we present the conditions for this product to be left cancellative, orthodox and finally left (right) inverse(s).

On state-closed representations of restricted wreath product of groups Gp,d = CpwrCd

Let Gp,d be the restricted wreath product CpwrCd where Cp is a cyclic group of order a prime p and Cd a free abelian group of finite rank d. We study the existence of faithful transitive state-closed (fsc) representations of Gp,d on the rooted m-ary tree for some finite m. The group G2,1, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for d≥2 there are no fsc representations of Gp,d on the p-adic tree. We describe all fsc representations of G=Gp,1 on the p-adic tree obtained via virtual endomorphisms, where the first level stabilizer of the image of G contains its commutator subgroup. Furthermore, for d≥2, we construct fsc representations of Gp,d on the p2-adic tree and exhibit concretely the representation of G2,2 on the 4-tree as a finite-state automaton group.

Distortion of wreath products in Thompson’s group F

The authors use geometric techniques to prove that the restricted wreath product F ≀ ℂ is a quasi-isometrically embedded subgroup of Thompson’s group F.

APPROXIMATION OF GEODESICS IN METABELIAN GROUPS

It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete [A. Myasnikov, V. Roman'kov, A. Ushakov and A. Vershik, The word and geodesic problems in free solvable groups, Trans. Amer. Math. Soc.362(9) (2010) 4655–4682] (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem. We show that the Geodesic Problem in the restricted wreath product of a finitely generated non-trivial group with a finitely generated abelian group containing ℤ2 is NP-hard and there exists a Polynomial Time Approximation Scheme for this problem. We also show that the Geodesic Problem in the restricted wreath product of two finitely generated non-trivial abelian groups is NP-hard if and only if the second abelian group contains ℤ2.

A measurable-group-theoretic solution to von Neumann’s problem

We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors.

A geometric classification of some solvable groups of homeomorphisms

We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of the generalized Thompson groups F_n. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the F_n. We also show that any standard restricted wreath product C wr T (of non-trivial groups) that embeds in PLo(I) or F_n must have T isomorphic with the integers.

The spectra of lamplighter groups and Cayley machines

We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr \(\mathbb{Z}\), with G a finite group, by calculating the Kesten—von Neumann—Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Żuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by an automaton coincides with the Kesten—von Neumann—Serre spectral measure.

TREE-WREATHING APPLIED TO GENERATION OF GROUPS BY FINITE AUTOMATA

We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n,ℤ), a canonical subgroup of finite index in GL(n,ℤ), then the restricted wreath product H wr B can be generated by finite synchronous automata on 0,1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0's is trivial. The uni-triangular group U = U(n,ℤ) is a subgroup of B(n,ℤ) and so, H wr U also can be generated by finite synchronous automata on 0,1.