Lida Chen, Jianchun Wu
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The $2$-closure $\overline{G}$ of a permutation group $G$ on $Ω$ is defined to be the largest permutation group on $Ω$, having the same orbits on $Ω\timesΩ$ as $G$. It is proved that if $G$ is supersolvable, then $\overline{G}$ can be found in polynomial time in $|Ω|$. As a byproduct of our technique, it is shown that the composition factors of $\overline{G}$ are cyclic or alternating of prime degree.
The pro-supersolvable topology on a free group: deciding denseness
本文研究自由群有限生成子群在 pro-超可解拓扑中的稠密性判定问题，证明其可判定，并指出该工作受有限可解群情形这一公开问题驱动。
We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.
Computing the Zariski closure of a finitely generated matrix group
本文提出了一种计算有限生成矩阵群扎里斯基闭包的算法，无需使用格罗伯纳基，并在OSCAR系统中实现了该算法。
It is shown that a group defined by forbidding all patterns of size s+1 that do not appear in a given self-similar group of tree automorphisms is the topological closure of a self-similar, countable, regular branch group, branching over its level s stabilizer. As an application, it is shown that there are no infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by forbidden patterns of size two.
The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.
In this paper, we work on the pro-nilpotent group topology of a free group. First we investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group topology. We present an algorithm for the calculation of the closure in the pro-nilpotent group topology of the product of finitely many finitely generated subgroups of a free group. Then we deduce that the nil-closure of a rational subset is computable. We also prove that the pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V, and J * G_nil are decidable.
Let $F$ be a finitely generated free group. We present an algorithm such that, given a subgroup $H\leqslant F$, decides whether $H$ is the fixed subgroup of some family of automorphisms, or family of endomorphisms of $F$ and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and geometric methods.
On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices
本文研究了矩阵有限生成群Zariski闭包的计算复杂度，提出新方法得到定义闭包的多项式次数界限，证明其可在初等时间内计算，并给出固定数域上线性代数群链长度的上界。
Computation of endo-fixed closures in free-abelian times free groups
本文研究自由阿贝尔乘自由群自同态不动子群的行为，提出一种结合组合与代数方法的算法，用于判定有限生成子群是否为某自同态族的不动子群并构造该族。
Schur's theorem and its relation to the closure properties of the non-abelian tensor product
本文研究了Noetherian群的Schur乘子生成性及非阿贝尔张量积的闭包性质，提出了Schur定理的新版本。
The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group.
Let W be an irreducible, finitely generated Coxeter group. The geometric representation provides an discrete embedding in the orthogonal group of the so-called Tits form. One can look at the representation modulo the kernel of this form; we give a new proof of the following result of Vinberg: if W is non-affine, then this representation remains faithful. Our proof uses relative Kazhdan Property (T). The following corollary was only known to hold when the Tits form is non-degenerate: the reduced C*-algebra of W is simple with a unique normalized trace. Some other corollaries are pointed out.
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 …
The pro-$k$-solvable topology on a free group
本文证明了自由群有限生成子群在pro-(元)交换和pro-${\bf V}$拓扑下闭包性质的可判定性，并给出了闭包基的有效构造及成员资格判定。
The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of $m,n$. We also study periodic points of endomorphisms of $Z^m\times F_n$, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of $Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real eleme…
Limits of dihedral groups
给出了二面体群极限的一个刻划.我们还描述了二面体群在标记群空间中在一定数量的生成元上的拓扑闭包。
On finitely generated Hausdorff spectra of groups acting on rooted trees
该研究证明有限生成正则分支群闭包的有限生成豪斯多夫谱为[0,1]，回答了Klopsch和Shalev的问题，推广了Abért-Virág结果，首次给出指定维数的显式子群，并定义良滤过解决相关问题。
Let G be a complete Kac-Moody group over a finite field. It is known that G possesses a BN-pair structure, all of whose parabolic subgroups are open in G. We show that, conversely, every open subgroup of G is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.
Groups of finite type: classification and structural properties
本文研究有限型群的性质，证明了一大类有限型群中有限生成、恰无限与强完全等价，改进相关算法并给出有限Mealy型自动机生成群闭包为有限型的充分条件，还提出同构判定算法并完成多类树上有限型群的同构分类。
A. K. Bousfield's $H\mathbb Z$-localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine's algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield $H\mathbb Z$-localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group $G$ to it's Bousfield $H\mathbb Z$-localiza…
Hausdorff dimension in a family of self-similar groups
对于每个素数p和一个在p上可逆的首一多项式f，我们定义了一个p元有根树的p进自同构群G_{p,f}。这些组在第一个Grigorchuk组之后建模，在该设置中，第一个Grigorchuk组是G_{2,x^2+x+1}组。我们证明了所构造的群是自相似的正则分支群。这使我们能够计算它们的闭包的豪斯多夫维数，提供了具体的例子（不使用随机方法）拓扑生成的闭子群的组的p-adic自同构与豪斯多夫维数任意接近1。我们利用分支性质给出了群的一个刻划，并由此得出所有定义的群都是群的分支。此外，我们证明了所有p-adic自同构的无穷群的Hausdorff维数都是正的和有理的，并给出了一个关于n约束群的豪斯多夫维数的一般公式.还讨论了进一步的“有限性”性质（顺从性，扭转和中间增长）。
We prove that Thompson's group $F$ has a subgroup $H$ such that the conjugacy problem in $H$ is undecidable and the membership problem in $H$ is easily decidable. The subgroup $H$ of $F$ is a closed subgroup of $F$. That is, every function in $F$ which is a piecewise-$H$ function belongs to $H$. Other interesting examples of closed subgroups of $F$ include Jones' subgroups $\overrightarrow{F}_n$ and Jones' $3$-colorable subgroup $\mathcal F$. By a recent result of the first author, all maximal subgroups of $F$ of infinite index are closed. In this paper we prove that if $K\leq F$ is finitely g…
Generic groups acting on regular trees
设T是k-正则树（k>2），A是它的自同构群。我们分析了A的一个泛子群Gamma。我们证明了Gamma是自由的，并在Gamma的闭包上建立了一个可分性：它要么是离散的，要么是紧的，要么在A中的指数至多为2。
Zariski-Closures of Linear Reflection Groups
本文给出了Vinberg意义下线性反射群Zariski稠密的充要条件，并证明了不可约直角Coxeter群在$\mathrm{SL}_n(\mathbb{Z})$中的虚拟Zariski稠密嵌入，解决了其Zariski稠密曲面子群的存在性问题。
We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a constructible group from the algebraic closure of A along cyclic subgroups. In particular, it follows that the algebraic closure of A is finitely generated, quasiconvex and hyperbolic. Suppose that G is free. Then the definable closure of A is a free factor of the algebraic closure of A and the rank of these groups is bounded by that of G. We prove that the algebraic c…
The homology groups introduced by A. Brumer can be used to establish a criterion ensuring that a profinite $\mathbb{F}_p[[G]]$-module of a pro-$p$ group $G$ has projective dimension $d<\infty$ (cf. Thm. A). This criterion yields a new characterization of free pro-$p$ groups (cf. Cor. B). Applied to a semi-direct factor $G\to\mathbb{Z}_p\to G$ isomorphic to $\mathbb{Z}_p$ which defines a non-trivial end in the sense of A.A. Korenev one concludes that the closure of the normal closure of the image of $σ$ is a free pro-$p$ subgroup (cf. Thm. C). From this result we will deduce a structure theorem…
We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).
Autostackability for finitely presented groups is a topological property of the Cayley graph combined with formal language theoretic restrictions, that implies solvability of the word problem. The class of autostackable groups is known to include all asynchronously automatic groups with respect to a prefix-closed normal form set, and all groups admitting finite complete rewriting systems. Although groups in the latter two classes all satisfy the homological finiteness condition $FP_\infty$, we show that the class of autostackable groups includes a group that is not of type $FP_3$. We also show…
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 conj…
Completely Syndetic Sets in Discrete Groups
本文研究离散群中的完全 syndetic 集，证明虚拟阿贝尔群不能划分为两个 CS 集，刻画了整数群的 CS 子集，并表明 CS 集密度可任意小。
Following the construction from `Degrees of growth of finitely generated groups and the theory of invariant means' we generalize the Grigorchuk's overgroup $\tilde{\mathcal{G}}$, studied in `On parabolic subgroups and Hecke algebras of some fractal groups' to the family $\{ \tilde{G}_ω, ω\in Ω= \{ 0, 1, 2 \}^\mathbb{N} \}$ of generalized Grigorchuk's overgroups. We consider these groups as 8-generated and describe the closure of this family in the space $\mathcal{M}_8$ of marked groups.
Density of finitely supported invariant measures for automorphisms of compact abelian groups
本文证明满足降链条件的紧可度量阿贝尔群自同构的有限支撑不变测度在弱-*拓扑下稠密，并利用specification性质变体得到Hilbert-Schmidt稳定性与Livshitz型定理等推论。
The Hausdorff dimension of the generalized Brunner-Sidki-Vieira Groups
本文计算了作用于 $m$-adic 树上的广义 Brunner-Sidki-Vieira 群闭包的 Hausdorff 维数，首次给出了 $m$-adic 自同构群中具有超越 Hausdorff 维数的自相似拓扑有限生成闭子群的实例。
Let F be a finitely generated field of characteristic zero and Γ<GL_n(F) a finitely generated subgroup. For an element g in Γ, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of Γ(but not on Γ).
Let $Γ_2\subseteq Γ_1$ be finitely generated subgroups of ${\rm GL}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let $\mathbb{G}_i$ be the Zariski-closure of $Γ_i$ in $({\rm GL}_{n_0})_{\mathbb{Q}}$, $\mathbb{G}_i^{\circ}$ be the Zariski-connected component of $\mathbb{G}_i$, and let $G_i$ be the closure of $Γ_i$ in $\prod_{p\nmid q_0}{\rm GL}_{n_0}(\mathbb{Z}_p)$. In this article we prove that, if $\mathbb{G}_1^{\circ}$ is the smallest closed normal subgroup of $\mathbb{G}_1^{\circ}$ which contains $\mathbb{G}_2^{\circ}$ and $Γ_2\curvearrowright G_2$ has spectral gap, then $Γ_1\curvearrowrigh…
On the complexity of the word problem of the R. Thompson group V
本研究分析Lehnert与Schweitzer的证明，得出R. Thompson群V及其有限生成子群的字问题的复杂性，确定其属于对数DCFL且在确定性多带图灵机上具二次时间复杂度。
Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that there exists a finitely generated group $Γ$ such that for any countable ordinal $α$ we have an equationally compact subgroup $H\subset Γ$ for which the Cantor-Bendixson rank of the orbit closure of $H$ equals to $α+2$. Then we give an explicite construction of continuum many equationally compact subgroups of $Γ$ such that the associated ergodic Bernoulli shi…
We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil-Rödl Theorem, the Ramsey property of partial orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered $S$-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the str…
It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: `Is every finitely generated perfect group the normal closure of one element?' We prove a weakened version of our question: there is no general procedure to pick a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose…
We initiate the study of a measurable analogue of small topological full groups that we call $\mathrm L^1$ full groups. These groups are endowed with a Polish group topology which admits a natural complete right invariant metric. We mostly focus on $\mathrm L^1$ full groups of measure-preserving $\mathbb Z$-actions which are actually a complete invariant of flip conjugacy. We prove that for ergodic actions the closure of the derived group is topologically simple although it can fail to be simple. We also show that the closure of the derived group is connected, and that for measure-preserving f…
Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Γ$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $\mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Γ$ determines the group $G$ upto isogeny over the algebraic closure of $K$.
We show that the class of finitely generated virtually free groups is precisely the class of demonstrable subgroups for R. Thompson's group $V$. The class of demonstrable groups for $V$ consists of all groups which can embed into $V$ with a natural dynamical behaviour in their induced actions on the Cantor space $\mathfrak{C}_2 := \left\{0,1\right\}^ω$. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while R. Thompson's group $V$ is a candidate as a universal $co\mathcal…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。