Sean Cleary
Wreath products such as Z wr Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z wr Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's metabelian group G. We find that Z wr Z is undistorted in F but is at least exponentially distorted in G.
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We characterize which permutational wreath products W^(X)\rtimes G are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X^2. On the one hand, this extends a result of G. Baumslag about standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characte…
Given a sequence of $(G_i)_{i \in \N}$ of finite transitive groups of degree $n_i$, let $W_\infty$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_\infty$ is (topologically) finitely generated if and only if $\prod_{i=1}^{\infty} (G_i/G_i')$ is finitely generated and the growth of the minimal number of generators of $G_i$ is bounded by $ d...n_1...n_{i-1}$ for a constant $d$. Moreover we give a criterion to decide whether $W_\infty$ is positively finitely generated.
We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid $M$ is said to preserve property $\mathcal{P}$ in direct products if, for any two $M$-acts $A$ and $B$, the direct product $A\times B$ has property $\mathcal{P}$ if and only if both $A$ and $B$ have property $\mathcal{P}$. It is proved that the monoids $M$ that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal $M$-act $M\times M$ is finitely generated (resp. fini…
For a finitely generated regular wreath product, the metric is known, but its computation can be an NP-complete problem. Also, it is not known for the nonregular case. In this article, a metric estimate is defined for regular wreath products which can be computed in polynomial time, based on the metrics of the factors. This estimate is then used to study the distortion of some natural subgroups of a wreath product. Finally, the metric estimate is generalized to the nonregular case.
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)…
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。
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 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 show that a wreath product of two finitely generated abelian groups is LERF. Consequently the free metabelian groups are LERF.
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。
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…
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 …
Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterat…
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 study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.
On linear divergence in finitely generated groups
本文研究了有限生成群中线性发散性的性质，证明了 wreath 积、halo 积以及 Diestel-Leader 图等结构具有线性发散性，并推广了相关论证方法。
We study infinitely iterated wreath products of finite permutation groups with respect to product actions. In particular, we prove that, for every non-empty class of finite simple groups $\mathcal{X}$, there exists a finitely generated hereditarily just infinite profinite group $W$ with composition factors in $\mathcal{X}$ such that any countably based profinite group with composition factors in $\mathcal{X}$ can be embedded into $W$. Additionally we investigate when infinitely iterated wreath products of finite simple groups with respect to product actions are co-Hopfian or non-co-Hopfian.
Property FW and wreath products of groups: a simple approach using Schreier graphs
本文利用 Schreier 图，为有限生成圈积具有性质 FW 的充要条件（因子群具性质 FW 且基集有限）提供了一个初等、直接且显式的证明。
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…
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。
The study of the poset of hyperbolic structures on a group $G$ was initiated in a paper by C.Abbott, S.Balasubramanya and D.Osin. However, this poset is still very far from being understood and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures: we give an example of a group with an uncountable chain of quasi-parabolic structures and p…
It is shown that membership in rational subsets of wreath products H \wr V with H a finite group and V a virtually free group is decidable. On the other hand, it is shown that there exists a fixed finitely generated submonoid in the wreath product Z \wr Z with an undecidable membership problem.
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.
We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.
We study generalisations 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 characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are …
A group has finite palindromic width if there exists $n$ such that every element can be expressed as a product of $n$ or fewer palindromic words. We show that if $G$ has finite palindromic width with respect to some generating set, then so does $G \wr \mathbb{Z}^{r}$. We also give a new, self-contained, proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.
In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.
We show that if $G$ and $H$ are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath $G \wr H$. We also prove an analogous result for coarse embeddings of wreath products. In the special case $G=\Z$, $H=\Z \wr \Z$ our result implies that the Hilbert compression exponent of $\Z\wr (\Z\wr\Z)$ is at least 1/4, answering a question posed as to whether it has positive Hilbert compression exponent.
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。
The degree of commutativity of wreath products with infinite cyclic top group
本文证明对形如$H \wr \langle t \rangle$（顶群无限循环）的受限圈积，无论生成集如何选择，其交换度恒为零，推广了Cox关于灯架群的结果，并给出了基群像密度为零的推广结论。
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.
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。
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。
In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.
On the quasi-isometric classification of permutational wreath products
本文研究满足非粗分离条件的置换圈积 $F\wr_{H/N}H$，基于 Genevois 与 Tessera 的工作给出其拟等距完整分类，并讨论双 Lipschitz 等价、缩放群及相关 halo 积。
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 study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generated subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group.
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…
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 give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for …
On the first-order genus of wreath products and their central extensions
证明 Z^m wr Z^n 与 Z 是正则双可解释的，因此一阶刚性；而 Z^2 wr Z 有 2^ℵ0 个初等等价但不同构的有限核中心扩张。
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…
Thompson's group $V$ has a rich variety of subgroups, containing all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of $V$.
Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $ζ(G,s) = \sum_{n=1}^{\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $ζ(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, a…
From any two median spaces $X,Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups $G,H$ and two (equivariant) coarse embeddings into median spaces $X,Y$, there exist a(n equivariant) coarse embedding $G\wr H \to X \circledast Y$. As an application, we prove that $$α_1(G \wr H) \geq \min(α_1(G),α_1(H))/2 \text{ for all finitely generated groups $G,H$,}$$ where $α_1(\cdot)$ denotes the $\ell^1$-compression. As an ot…
It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. For each positive rational number $s$ we construct pairs of finitely presented groups $H\subset G$ such that the distortion of $H$ in $G$ is $\simeq n^s$. And for each $s\ge 1$ we also construct finitely presented groups whose isodiametric function is $\simeq n^s$.
Isoperimetric inequalities in finitely generated groups
本文研究了有限生成群的Dehn谱，计算了多种群的Dehn谱，并证明了存在不可数多个非准等距的有限生成群。
We study the isoperimetric profiles of certain families of finitely generated groups defined via marked Schreier graphs and permutation wreath products. The groups we study are among the "simplest" examples within a much larger class of groups, all defined via marked Schreier graphs and/or action on rooted trees, which includes such examples as the long range group, Grigorchuck group and the basillica group. The highly non-linear structure of these groups make them both interesting and difficult to study. Because of the relative simplicity of the Schreier graphs that define the groups we study…
Continuity of asymptotic entropy on wreath products
本文证明了在缠绕积上非退化概率测度的渐近熵作为步分布函数的连续性，并扩展了该结果到新的群类。
Diophantine Problem in Some Metabelian Groups
该研究证明了Baumslag-Solitar群$BS(1,k)$与有限生成阿贝尔群和无限循环群的圈积中，二次方程的丢番图问题具有可判定性，拓展了群论中丢番图问题的可判定范围。
In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product $\mathbb Z\wr (\mathbb Z\times \mathbb Z)$. We also show that subsemimodule membership is undecidable for finite rank free $(\mathbb Z\times \mathbb Z)$-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.
We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.
Wreath products of groups acting with bounded orbits
本文证明了对于多种度量空间子范畴，圈积群具有性质BS当且仅当因子群均具此性质且作用集合有限，推广了关于性质FH、FW和Bergman性质的结果。
Reidemeister classes, wreath products and solvability
本文研究了限制圈积中的Reidemeister类，在自同构阶数和Reidemeister类数目有限的条件下证明了猜想TBFT_f，并证明了满足该条件的有限生成剩余有限群具有可解有限性（猜想R）。
We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontriv…
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。
We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\ell^Φ(G)$, where $Φ$ is an $N$-function of class $Δ_2(0)\cap \nabla_2(0)$. In development of ideas of Puls and Martin--Valette, for a finitely generated group $G$, we introduce the discrete $Φ$-Laplacian and prove a theorem on the decomposition of the space of $Φ$-Dirichlet finite functions into the direct sum of the spaces of $Φ$-harmonic functions and $\ell^Φ(G)$ (with an appropriate factorization). We also prove that if a finitely generated group $G$ has a finitely generated infinite a…
Dead ends on wreath products and lamplighter groups
本文证明任意有限群与有限生成群的圈积（灯夫群）存在无界深度的标准生成集，并在阿贝尔基群情形推广到所有标准生成集；对自由积基群，用几何常数刻画无界深度，并发现与一维情形的差异。
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