Alex C. Dantas, Said N. Sidki
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.
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We present a survey of our recent research on varieties, generated by wreath products of groups. In particular, the full classification of all cases, when the (cartesian or direct) wreath product of any abelian groups $A$ and $B$ generates the product variety $\var{A} \var{B}$, is given. The analog of this is given for sets of abelian groups. We also present new, unpublished research on cases when the similar question is considered for non-abelian, finite groups.
The Diophantine problem in iterated wreath products of free abelian groups is undecidable
本文证明有限秩非平凡自由阿贝尔群（k>1）的迭代限制圈积中的丢番图问题不可判定，即无算法判定该群中有限系数群方程组是否有解。
Generalizing results of Higman and Houghton on varieties generated by wreath products of finite cycles, we prove that the (direct or cartesian) wreath product of arbitrary abelian groups $A$ and $B$ generates the product variety $var (A) \cdot var (B)$ if and only if one of the groups $A$ and $B$ is not of finite exponent, or if $A$ and $B$ are of finite exponents $m$ and $n$ respectively and for all primes $p$ dividing both $m$ and $n$, the factors $B[p^k]/B[p^{k-1}]$ are infinite, where $B[s]=\langle b\in B|\,b^{s}=1 \rangle$ and where $p^k$ is the highest power of $p$ dividing $n$.
Bounding conjugacy depth functions for wreath products of finitely generated abelian groups
本文研究了阿贝尔群圈积的共轭可分性渐近行为，完全刻画了lamplighter群的渐近类并给出了广义lamplighter群及无限基群情形的指数或超指数上下界，进而为非阿贝尔作用群的各种群圈积的共轭深度函数提供了下界。
On the residual finiteness of the non-abelian exterior square of wreath products and the Grigorchuk group
本文给出圈积及其变体的非阿贝尔外平方剩余有限的充分条件，并证明Grigorchuk群的外平方与张量积剩余有限，同时给出圈积非上同调好的条件。
We show that a wreath product of two finitely generated abelian groups is LERF. Consequently the free metabelian groups are LERF.
We present a general criterion under which the equality var(A Wr B) = var(A) var(B) holds for finite groups A and B. This continues our previous research on varieties, generated by wreath products of abelian groups, and generalizes some existing results in this direction in literature. The classification is based on criterion of A.L. Shmel'kin for product varieties of groups and on technics developed by R. Burns et al. on critical groups in nilpotent-by-abelian varieties.
We show that the wreath product $G \wr \mathbb{Z}^n$ of any finitely generated group $G$ with $\mathbb{Z}^n$ has finite palindromic width. We also show that $C \wr A$ has finite palindromic width if $C$ has finite commutator width and $A$ is a finitely generated infinite abelian group. Further we prove that if $H$ is a non-abelian group with finite palindromic width and $G$ any finitely generated group, then every element of the subgroup $G' \wr H$ can be expressed as a product of uniformly boundedly many palindromes. From this we obtain that $P \wr H$ has finite palindromic width if $P$ is a …
We find a necessary condition for the embedding of a central extension of a group $G$ with elementary abelian kernel into the wreath product that corresponds to a permutation action of $G$. The proof uses purely group-theoretic methods.
On the Self-Similarity of Permutational Wreath Products and Their Embedding into Finitely Presented Simple Groups
该研究探讨置换圈积的自相似性，证明特定条件下Scott–Röver–Nekrashevych群的有限表现性与几乎单性，及\boldsymbol{\boldsymbol{Z}^d \rtimes_X G}嵌入有限表现单群，为Boone–Higman猜想提供新群族。
We classify certain cases when the wreath products of distinct pairs of groups generate the same variety. This allows us to investigate the subvarieties of some nilpotent-by-abelian product varieties ${\mathfrak U}{\mathfrak V}$ with the help of wreath products of groups. In particular, using wreath products we find such subvarieties in nilpotent-by-abelian ${\mathfrak U}{\mathfrak V}$, which have the same nilpotency class, the same length of solubility, and the same exponent, but which still are distinct subvarieties. Obtained classification strengthens our recent work on varieties generated …
Let $A$ be a nilpotent $p$-group of finite exponent, and $B$ be an abelian $p$-groups of finite exponent. Then the wreath product $A {\rm Wr} B$ generates the variety ${\rm var}(A) {\rm var}(B)$ if and only if the group $B$ contains a subgroup isomorphic to the direct product $C_{p^v}^\infty$ of at least countably many copies of the cyclic group $C_{p^v}$ of order $p^v = \exp{(B)}$. The obtained theorem continues our previous study of cases when ${\rm var}(A {\rm Wr} B ) = {\rm var}(A){\rm var}(B)$ holds for some other classes of groups $A$ and $B$ (abelian groups, finite groups, etc.).
Orientable quadratic equations in wreath products
本文研究有限生成阿贝尔群圈积 $A\wr B$ 中可定向二次方程的求解复杂度，并基于亏格等特征对问题计算困难或可行的情形进行了分类。
We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C \wr F$, where $C$ is a finite group and $F$ a non-abelian free group.
The RP-property of Fel'shtyn and Troitsky is proved for wreath products of finitely generated Abelian groups with the group of integers. Such wreath products become the first known example of finitely generated RP-groups being not almost polycyclic.
We show that the conjugacy problem in a wreath product $A \wr B$ is uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the factors $A$ and $B$ and the power problem in $B$. If $B$ is torsion free, the power problem for $B$ can be replaced by the slightly weaker cyclic submonoid membership problem for $B$. Moreover, if $A$ is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-$\mathsf{AC}^0$-many-one-reducible to the conjugacy problem in $A \wr B$. Furthermore, under certain natural conditions, we give a uniform $\mathsf{TC}^0$ Turing reduction f…
We characterize permutational wreath products with Property (FA). For instance, the standard wreath product A wr B of two nontrivial countable groups A,B, has Property (FA) if and only if B has Property (FA) and A is a finitely generated group with finite abelianisation. We also prove an analogous result for hereditary Property (FA). On the other hand, we prove that many wreath products with hereditary Property (FA) are not quotients of finitely presented groups with the same property.
We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product $G=T^k{:}P$ acting on its base group $Ω=T^k$, where $T$ is some non-abelian simple group and $P$ is some group acting transitively on $\boldsymbol{k}=\{1,\ldots,k\}$ with $k\geq 2$. We prove that if $G$ is primitive on $Ω$ and $P$ is quasiprimitive on $\boldsymbol{k}$,…
We present a full description of the Bieri-Neumann-Strebel invariant of restricted permutational wreath products of groups. We also give partial results about the 2-dimensional homotopical invariant of such groups. These results may be turned into a full picture of these invariants when the abelianization of the basis group is infinite. We apply these descriptions to the study of the Reidemeister number of automorphisms of wreath products in some specific cases.
Hopfian combinatorial wreath products
该研究探讨了组合缠绕积的可逆性条件，通过构造具体例子展示了不可逆情况，并分析了自动群的结构。
We study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated)…
Let $G$ be a finite group, $A$ a finite abelian group. Each homomorphism $ϕ:G\to A\wr S_n$ induces a homomorphism $\barϕ:G\to A$ in a natural way. We show that as $ϕ$ is chosen randomly, then the distribution of $\barϕ$ is close to uniform. As application we prove a conjecture of T. Müller on the number of homomorphisms from a finite group into Weyl groups of type $D_n$.
$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products
该研究给出限制 wreath 积为 $\boldsymbol{\textit{C}}$-遗传共轭可分群的充要条件，证明 Grigorchuk 群是 2-遗传共轭可分群，构造了无穷多类非多循环遗传共轭可分群。
A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be pre…
Gelfand pairs involving the wreath product of finite abelian groups with symmetric groups
本文推广了对称群的Gelfand对理论，证明了对于有限群G，其与对称群的圈积构成Gelfand对的充分必要条件是G为阿贝尔群。
We suggest a criterion under which for a nilpotent group of finite exponent $A$ and for an abelian group $B$ the variety $var(A \,Wr\, B)$ generated by their wreath product $A \,Wr\, B$ is equal to the product of varieties $var(A)$ and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only if either the group $B$ is not of some non-zero exponent; or if $B$ is of a non-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, $d$ is the largest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct…
Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely generated if and only if the profinite abelian group $\prod_{n\geq 1} G_n/G'_n$ is topologically finitely generated. As a corollary, for a finite transitive group $G$ the minimal number of generators of the wreath power $G\wr...\wr G\wr G$ ($n$ times) is bounded if $G$ is perfect, and grows linearly if $G$ is non-perfect. As a by-product we construct a finitely generate…
Kernel--wreath constructions and infinite families of finite simple skew braces
本文提出 kernel--wreath 构造，从单斜括号生成无限族有限单斜括号，保持可解性，并应用于阶 12 及 Byott 族。
Let $Λ$ be an ordered abelian group, $\mathrm{Aut}^+(Λ)$ the group of order-preserving automorphisms of $Λ$, $G$ a group and $α:G\to\mathrm{Aut}^+(Λ)$ a homomorphism. An $α$-affine action of $G$ on a $Λ$-tree $X$ is one that satisfies $d(gx,gy)=α_gd(x,y)$ ($x,y\in X$, $g\in G$). We consider classes of groups that admit a free, rigid, affine action in the case where $X=Λ$. Such groups form a much larger class than in the isometric case. We show in particular that unitriangular groups $\mathrm{UT}(n,\mathbb{R})$ and groups $T^*(n,\mathbb{R})$ of upper triangular matrices over $\mathbb{R}$ with p…
A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in th…
Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups.
Permutational wreath pullbacks and framed braid-type groups
本文引入置换 wreath 拉回结构，研究其中心和交换化，并探讨其作为经典 wreath 产品拉回的解释，同时建立有限生成阿贝尔群的条件，以继承 R_∞ 性质。
Graph and wreath products of cellular automata
本文证明了双侧全移位自同构群的子群在可数图积下封闭，引入无 $A$-消除的群作用概念，并证明有限阿贝尔群与元胞自动机群的圈积可嵌入全移位自同构群，同时将结论推广至单侧情形。
In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group $G$, knapsack (as well as the related sub…
Equations in wreath products
该研究综述群圈积中方程的可解性，证明二次丢番图问题在阿贝尔群圈积及Baumslag有限表现亚阿贝尔群中可解，同时探讨了换位子宽度的相关问题。
We give a partial solution a question of Grigorchuk, Nekrashevych, Sushchanskii and Šunik by giving an algorithm to test whether a finite state element of an infinite iterated (permutational) wreath product $\hat G = \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr >...$ of cyclic groups of order $n$ acts spherically transitively. We can also decide whether two finite state spherically transitive elements of $\hat G$ are conjugate. For general infinite iterated wreath products, an algorithm is presented to determine whether two finite state automorphisms have the same im…
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 pr…
A note on morphisms to wreath products
该研究探讨了有限生成群到 wreath 产品映射的性质，证明了映射核包含非交换自由子群，并扩展了 SQ-通用性的结论，同时揭示了与 Kaplansky 猜想的联系。
Let $K$ be field of characteristic 2 and let $G$ be a finite non-abelian 2-group with the cyclic derived subgroup $G'$, and there exists a central element $z$ of order 2 in $Z(G) \backslash G'$. We prove that the unit group of the group algebra $KG$ possesses a section isomorphic to the wreath product of a group of order 2 with the derived subgroup of the group $G$, giving for such groups a positive answer to the question of A. Shalev.
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 lev…
Transfinite hypercentral iterated wreath product of integral domains
本文研究特征0整环上的迭代圈积，通过数值多项式构造超限超中心群，并建立其与Lie环的结构对应及正规化子链刻画。
Random Schrödinger operators and convolution on wreath products
本文建立了随机 Schrödinger 算子与圈积上确定性卷积算子间的谱对应关系，并据此获得了卷积绝对连续性判据、Lifschitz 尾估计及 Green 函数二阶矩的精确公式。
The Conjugacy Problem in Wreath Products
研究花环积中共轭问题的可解性条件。通过指出简·马修斯结论的不足，给出新的附加条件。还给出了置换受限花环积中共轭问题可解的充要条件，涉及群\(A\)、\(B\)的共轭问题、幂问题、阶问题等多种条件。
Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces X,Y,Z and derive a condition, called the (delta-polynomial) path lifting property, such that coarse embeddability of X,Y and Z implies coarse embeddability of X\wr_Z Y. We also give bounds on the compression of X\wr_Z Y in terms of delta and the compressions of X,Y and Z. Next, we investigate the stability of the property of admitting a box space which coarsely embeds i…
We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $(2+ε)$-moment. This result is in contrast with classical examples of abelian groups, where the displacement after $n$ steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised $n$-step displacement admits a density whose support is $[0,\infty)$. We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not ho…
The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge 0$ with $g_1^{x_1}\cdots g_n^{x_n}=g$. We study the knapsack problem for wreath products $G\wr H$ of groups $G$ and $H$. Our main result is a characterization of those wreath products $G\wr H$ for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors $G$ and $H$. To this end, we introduc…
Consider the wreath product $Γ= F\wr \mathrm{F_n} = \bigoplus_{\mathrm{F_n}}F\rtimes\mathrm{F_n}$, with $F$ a finite group and $\mathrm{F_n}$ the free group on $n$ generators. We study the Baum-Connes conjecture for this group. Our aim is to explicitly describe the Baum-Connes assembly map for $F\wr \mathrm{F_n}$. To this end, we compute the topological and the analytical K-groups and exhibit their generators. Moreover, we present a concrete 2-dimensional model for $\underline{E} Γ$. As a result of our K-theoretic computations, we obtain that $\mathrm K_0(\mathrm C^*_{\mathrm r}(Γ))$ is the fr…
Hopfian wreath products and the stable finiteness conjecture
本文研究阿贝尔基群环积的 Hopf 性质，证明 Kaplansky 稳定有限性猜想等价于所有有限生成阿贝尔群与有限生成 Hopfian 群的环积均为 Hopfian，并进一步将其与加法细胞自动机的 Gottschalk 满射性猜想联系起来。
It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete (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 $Z^2$ 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-trivia…
Random walks on wreath products and spectral gaps for coloured interchange processes
研究 wreath 积上的群值彩色交换过程，通过表示论刻画其谱隙，在阿贝尔基群下关联离散薛定谔算子，并分类确定谱隙的极小不可约表示族。
Spinning switches on a wreath product
本文将旋转桌开关谜题推广为通过圈积组合的任意有限群，分类了可解谜题的圈积族，为阿贝尔群完全分类、为 $p$-群构造必胜策略，并给出含 monster group $M$ 的非阿贝尔群新示例。
On wreath product occurring as subgroup of automata group
本文证明若G为自同构群子群，则对任意有限生成阿贝尔群A，圈积A≀G仍为自同构群子群，并解决Kourovka笔记中关于Z≀(Z≀Z)为双字母自同构群子群的问题。
We show that the Magnus embedding, which embeds the free solvable group of rank r and degree d into the wreath product of the free abelian group of rank r with the free solvable group of rank r and degree d-1, is a quasi-isometry.
The complexity of knapsack problems in wreath products
本文研究群的 wreath 积中幂字问题和背包问题的复杂度，证明了幂零群 wreath 积等的幂字问题属 $\mathsf{TC}^0$，自由可解群背包问题为 $\mathsf{NP}$-完全，部分非可解群 wreath 积问题复杂度更高。
In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $\mathrm{cl}_{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(\mathbb{Z}\wr Γ, \bigoplus_Γ\mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $Γ$ is not locally cyclic, the ordinary commutator length $\mathrm{cl}_G$ does not coincide with $\mathrm{cl}_{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if …
A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a $CI$-Schur ring. By using this condition we reprove in a short way known results on the $CI$-property for decomposable Schur rings over an elementary abelian group of rank at most $5$.
Wreath products in the automorphism group of a full shift
本文证明，若非平凡全转移的自同构群子群H在有限支撑点上具有自由轨道，则对任意有限生成阿贝尔群A，抽象群A≀H可嵌入自同构群，并给出Z≀Z等新嵌入实例。
We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.
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.
We prove that all finitely generated free metabelian groups are permutation stable. This partially answers to the question asked by Levit and Lubotzky whether all finitely generated metabelian groups are permutation stable. Our proof extends the range of application of Levit-Lubotzky's method, which is used to show permutation stability of permutational wreath products of finitely generated abelian groups, to non-split and non-permutational metabelian groups.
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