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[1] arXiv:2507.03100 [中文pdf, pdf, html, 其他]

标题： 关于特征次数和代码数的平方自由最大公约数的群 显示英文标题

标题： On groups with square-free gcd of character degree and codegree

Karam Aldahleh, Alan Kappler, Neil Makur, Yong Yang

评论： 10页

主题： 群论 (math.GR)

设$G$为一个有限群，$\chi$为$G$的一个不可约特征标。$\chi$的代码度定义为$\chi^c(1) =\frac{|G: \ker\chi|}{\chi(1)}$。 在高、王和陈的论文\cite{GWC24}中，证明了$G$不能满足$\gcd(\chi(1),\chi^c(1))$对所有$\chi\in\Irr(G)^\#$为素数的条件。 我们通过解决钱国华的一个未解问题——关于字符码度的问题，对该定理进行了推广。 在钱的综述文章\cite{Qiancodegree}中，他询问了具有平方自由$\gcd$的不可解有限群的结构。 特别是，我们证明如果$G$满足对于每个不可约特征标$\chi$，$\gcd(\chi(1),\chi^c(1))$是平方自由的，那么$G/\text{Sol}(G)$同构于一个特定列表中的几乎单群之一。 显示英文摘要

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. The codegree of $\chi$ is defined as $\chi^c(1) =\frac{|G: \ker\chi|}{\chi(1)}$. In a paper by Gao, Wang, and Chen \cite{GWC24}, it was shown that $G$ cannot satisfy the condition that $\gcd(\chi(1),\chi^c(1))$ is prime for all $\chi\in\Irr(G)^\#$. We generalize this theorem by solving one of Guohua Qian's unsolved problems on character codegrees. In Qian's survey article \cite{Qiancodegree}, he inquires about the structure of non-solvable finite groups with square-free $\gcd$ instead. In particular, we prove that if $G$ is such that $\gcd(\chi(1),\chi^c(1))$ is square-free for every irreducible character $\chi$, then $G/\text{Sol}(G)$ is isomorphic to one among a particular list of almost simple groups.

[2] arXiv:2507.03667 [中文pdf, pdf, html, 其他]

标题： 非可定向的具有负素数幂欧拉特征的正则映射 显示英文标题

标题： Non-orientable regular maps with negative prime-power Euler characteristic

Marston Conder, Nick Gill, Jozef Širáň

评论： 42页

主题： 群论 (math.GR) ; 组合数学 (math.CO)

在本文中，我们对欧拉特征为$-r^d$的曲面上的所有正则映射进行了分类，其中某些奇素数$r$和整数$d\ge 1$。这样的映射必定是不可定向的，当$d = 1$或$2$时的情况之前已经处理过。 这种分类自然地分为三部分，基于映射的自同构群$G$的性质，特别是其商群$G/O(G)$的结构，其中$O(G)$是$G$中最大奇数阶的正规子群。 事实上，$G/O(G)$与一个$2$-群同构（在这种情况下，$G$是可解的），或者为$\textrm{PSL}(2,q)$或$\textrm{PGL}(2,q)$，其中$q$是一个奇素数幂。 结果是一组$18$个非空的正则映射族，对相关参数有特定条件。 显示英文摘要

In this paper we provide a classification of all regular maps on surfaces of Euler characteristic $-r^d$ for some odd prime $r$ and integer $d\ge 1$. Such maps are necessarily non-orientable, and the cases where $d = 1$ or $2$ have been dealt with previously. This classification splits naturally into three parts, based on the nature of the automorphism group $G$ of the map, and particularly the structure of its quotient $G/O(G)$ where $O(G)$ is the largest normal subgroup of $G$ of odd order. In fact $G/O(G)$ is isomorphic to either a $2$-group (in which case $G$ is soluble), or $\textrm{PSL}(2,q)$ or $\textrm{PGL}(2,q)$ where $q$ is an odd prime power. The result is a collection of $18$ non-empty families of regular maps, with conditions on the associated parameters.

[3] arXiv:2507.04065 [中文pdf, pdf, html, 其他]

标题： 不连通李群中的通用无限生成、固定点贫乏表示和紧元丰富性 显示英文标题

标题： Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups

Alexandru Chirvasitu

评论： 11页 + 参考文献

主题： 群论 (math.GR) ; 代数拓扑 (math.AT) ; 一般拓扑 (math.GN) ; 表示理论 (math.RT)

半直积$\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$在紧群作用于连通、单连通可解李群上时，其紧元素的集合稠密当且仅当在$\mathbb{L}$上自由不动点操作的$s\in \mathbb{K}$构成一个稠密集合。 这（以及若干其他等价的描述）扩展了吴针对连通李$\mathbb{K}$的类似结果，同时也提供了大量几乎连通李群$\mathbb{G}$的例子，这些群没有稠密的紧元素集合，尽管它们的单位成分$\mathbb{G}_0$有。 这修正了先前关于该主题的文献，声称 $\mathbb{G}$ 和 $\mathbb{G}_0$ 具有等价的性质。 在相关讨论中，我们刻画了那些连通李群， $\mathbb{G}$ 具有大量 $d$ 元组，生成稠密子群， $\Gamma\le \mathbb{G}$，对于这些李群，导出子群 $\Gamma^{(1)}$ 不是有限生成的：$\mathbb{G}$ 必须是非平凡拓扑完美的，或者具有非幂零最大可解商。 显示英文摘要

The semidirect product $\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$ attached to a compact-group action on a connected, simply-connected solvable Lie group has a dense set of compact elements precisely when the $s\in \mathbb{K}$ operating on $\mathbb{L}$ fixed-point-freely constitute a dense set. This (along with a number of alternative equivalent characterizations) extends the Wu's analogous result for connected Lie $\mathbb{K}$, and also provides ample supplies of examples of almost-connected Lie groups $\mathbb{G}$ which do not have dense sets of compact elements, even though their identity components $\mathbb{G}_0$ do. This corrects prior literature on the subject, claiming the property equivalent for $\mathbb{G}$ and $\mathbb{G}_0$. In a related discussion we characterize those connected Lie groups $\mathbb{G}$ with large sets of $d$-tuples generating dense subgroups $\Gamma\le \mathbb{G}$ for which the derived subgroup $\Gamma^{(1)}$ fails to be finitely-generated: $\mathbb{G}$ must either be non-trivial topologically perfect or have non-nilpotent maximal solvable quotient.

[4] arXiv:2507.04120 [中文pdf, pdf, html, 其他]

标题： 论自由群和自由p-群的共轭子群 显示英文标题

标题： On commensurators of free groups and free pro-p groups

Yiftach Barnea, Mikhail Ershov, Adrien Le Boudec, Colin D. Reid, Matteo Vannacci, Thomas Weigel

评论： 56页

主题： 群论 (math.GR)

我们研究自由群和自由质$p$群的共轭子群，以及这些群的某些子群。 我们证明了一个有限秩$F$的非交换自由群的共轭子群$Comm(F)$并不是虚拟简单的，回答了Lubotzky的问题。 另一方面，我们展示了一族容易定义的$Comm(F)$的有限生成子群，并表明这个族中的一些群是简单的。 对于素数$p$，我们还考虑 p-共轭子群$Comm_p(F)$，这是将$F$视为具有拟$p$拓扑的群时的共轭子群。 与$Comm(F)$相比，我们证明$Comm_p(F)$有一个指数至多为 2 的简单子群。 此外，虽然$Comm(F)$的同构类不依赖于$F$的秩，但我们证明了$Comm_p(F)$的同构类依赖于$F$的秩，并确定了确切的依赖关系。 如果$\mathbf F$是$F$的 pro-$p$完成（这是一个自由 pro-$p$群），则$Comm(\mathbf F)$是一个包含$\mathbf F$作为开子群的完全不连通局部紧（tdlc）群。 我们使用$Comm_p(F)$来构造一个抽象简单的子群，该子群包含$Comm(\mathbf F)$以及一个包含$\mathbf F$的非离散 tdlc 群族，这些群是紧生成的且简单的。 显示英文摘要

We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.

[5] arXiv:2507.04347 [中文pdf, pdf, html, 其他]

标题： Higman嵌入定理的显式算法 显示英文标题

标题： An explicit algorithm for the Higman Embedding Theorem

V.H. Mikaelian

主题： 群论 (math.GR)

我们提出了一种算法，对于任何递归群$G$，由其有效枚举的生成元和递归枚举的关系给出，该算法输出一个显式的嵌入，将$G$嵌入到一个直接由其生成元和定义关系写出的有限表示群中。 这是著名的希格曼嵌入定理的显式类比，该定理指出，一个有限生成群$G$可以嵌入到一个有限表示群中当且仅当$G$是递归的。 构造出的有限表示群甚至可以被选择为$2$生成的。 该算法已经应用于例如有理数加法群$\mathbb Q$，它是递归的。 关于$\mathbb Q$显式嵌入到有限表示群中的问题在文献中已被约翰逊、德·拉哈普、布里德森等人提到过。 所建议的方法也可以用于解决其他一些递归群的嵌入问题。 嵌入算法是使用传统自由构造及其修改构建的，包括带有合并的自由积、HNN-扩张以及某些辅助$*$-构造。 我们还分析了原始Higman嵌入的步骤，以指出其中哪些部分是显式的或不是显式的。 显示英文摘要

We propose an algorithm which for any recursive group $G$, given by its effectively enumerable generators and recursively enumerable relations, outputs an explicit embedding of $G$ into a finitely presented group directly written by its generators and defining relations. This is the explicit analogue of the remarkable Higman Embedding Theorem stating that a finitely generated group $G$ is embeddable into a finitely presented group if and only if $G$ is recursive. The constructed finitely presented group can even be chosen to be $2$-generator. This algorithm has already been applied, for example, to the additive group of rational numbers $\mathbb Q$, which is recursive. The question on explicit embedding of $\mathbb Q$ into a finitely presented group was mentioned in the literature by Johnson, De la Harpe, Bridson and others. The suggested method can be used to solve the problem of embeddings for some other recursive groups, also. The embedding algorithm is built using conventional free constructions and their modifications, including free products with amalgamation, HNN-extensions, and certain auxiliary $*$-constructions. We also analyze the steps of original Higman embedding to indicate which of its parts are or are not explicit.

[6] arXiv:2507.04486 [中文pdf, pdf, 其他]

标题： 独异点的扭曲乘积 显示英文标题

标题： Twisted products of monoids

James East, Robert D. Gray, P.A. Azeef Muhammed, Nik Ruškuc

主题： 群论 (math.GR) ; 环与代数 (math.RA)

一个独异点$S$的扭曲是一个满足等式$\Phi(a,b) + \Phi(ab,c) = \Phi(a,bc) + \Phi(b,c)$的映射$\Phi:S\times S\to\mathbb{N}$。 与一个加法交换独异点$M$和一个固定的$q\in M$一起，这产生了一个所谓的扭曲乘积$M\times_\Phi^qS$，其底层集为$M\times S$且乘法为$(i,a)(j,b) = (i+j+\Phi(a,b)q,ab)$。 这种构造在特殊情况下出现过，其中$M$是$\mathbb{N}$或$\mathbb{Z}$在加法下，$S$是一个图示独异点（例如，划分、Brauer 或 Temperley-Lieb），而$\Phi$是连接图示中浮动成分的数量。 在本文中，我们确定了一种特殊的“紧”扭曲，并给出了由此产生的扭曲乘积的详细结构描述。 这包括表征格林关系、(von Neumann) 正则元素、幂等元、双序集、极大子群、Schützenberger 群等。 我们还考虑了一些例子，包括几个似乎新的例子，这些例子以线性代数中Sylvester秩不等式的某些推广为起点。 显示英文摘要

A twisting of a monoid $S$ is a map $\Phi:S\times S\to\mathbb{N}$ satisfying the identity $\Phi(a,b) + \Phi(ab,c) = \Phi(a,bc) + \Phi(b,c)$. Together with an additive commutative monoid $M$, and a fixed $q\in M$, this gives rise a so-called twisted product $M\times_\Phi^qS$, which has underlying set $M\times S$ and multiplication $(i,a)(j,b) = (i+j+\Phi(a,b)q,ab)$. This construction has appeared in the special cases where $M$ is $\mathbb{N}$ or $\mathbb{Z}$ under addition, $S$ is a diagram monoid (e.g.~partition, Brauer or Temperley-Lieb), and $\Phi$ counts floating components in concatenated diagrams. In this paper we identify a special kind of `tight' twisting, and give a thorough structural description of the resulting twisted products. This involves characterising Green's relations, (von Neumann) regular elements, idempotents, biordered sets, maximal subgroups, Sch\"{u}tzenberger groups, and more. We also consider a number of examples, including several apparently new ones, which take as their starting point certain generalisations of Sylvester's rank inequality from linear algebra.

[7] arXiv:2507.04519 [中文pdf, pdf, html, 其他]

标题： 局部各向同性Steinberg群II。 Schur乘子 显示英文标题

标题： Locally isotropic Steinberg groups II. Schur multipliers

Egor Voronetsky

评论： 初稿

主题： 群论 (math.GR) ; K理论与同调 (math.KT)

我们计算局部各向同性Steinberg群以及所有根分级Steinberg群的Schur乘子，其根系的秩至少为$ 3 $（不包括类型$ \mathsf H_3 $和$ \mathsf H_4 $）。 作为应用，我们证明局部各向同性Steinberg群作为抽象群是定义良好的。 显示英文摘要

We compute Schur multipliers of locally isotropic Steinberg groups and of all root graded Steinberg groups with root systems of rank at least $ 3 $ (excluding the types $ \mathsf H_3 $ and $ \mathsf H_4 $). As an application, we show that locally isotropic Steinberg groups are well defined as abstract groups.

[8] arXiv:2507.04577 [中文pdf, pdf, html, 其他]

标题： 二阶积分同调群的偶数阿廷群 显示英文标题

标题： The second integral homology of even Artin groups

Toshiyuki Akita

评论： 将出现在九州大学数学杂志上

主题： 群论 (math.GR) ; 代数拓扑 (math.AT)

在本文中，我们使用霍普夫公式计算了偶数阿廷群的第二整数同调群。然后，我们将我们的结果应用于偶数阿tin群上乘积和庞特里亚金积的计算，以及偶数考克斯特群的第二整数同调群。 显示英文摘要

In this paper, we compute the second integral homology of even Artin groups using Hopf's formula. We then apply our results to the computation of cup products and Pontryagin products on even Artin groups, as well as to the second integral homology of even Coxeter groups.

[9] arXiv:2507.04819 [中文pdf, pdf, html, 其他]

标题： 特殊和单关系式独异点的双侧同调性质 显示英文标题

标题： Two-sided homological properties of special and one-relator monoids

Robert D. Gray, Benjamin Steinberg

评论： 22页，1图

主题： 群论 (math.GR) ; 环与代数 (math.RA)

如果每个定义关系的右边都等于1，则称为特殊独异点表示。我们证明了有关由特殊表示定义的独异点的双向同调有限性质与其单位群的同调有限性质之间的结果。具体来说，我们证明了当其单位群属于类型$\mathrm{FP}_n$时，该独异点具有双向$\mathrm{FP}_n$的同调有限性质。我们还得到了有关独异点的Hochschild上同调维数与它的单位群的上同调维数之间的关系的结果。特别是，我们证明了独异点的Hochschild上同调维数不超过2和其单位群的上同调维数的最大值。我们将这些结果应用于证明一种类似于Lyndon恒等式的定理，用于形式为$\langle A \mid r=1 \rangle$的单关系独异点的双向同调。特别是，我们证明了所有这样的独异点都是双向$\mathrm{FP}_\infty$类型的。此外，我们证明了如果$r$不是适当幂，则单关系独异点的Hochschild上同调维数最多为$2$，而如果$r$是适当幂，则其Hochschild上同调维数为无限。 对于非特殊的一关系独子半群，其定义关系为$u=v$，我们证明如果不存在非空字$r$使得$u,v \in A^*r \cap r A^*$，则$M$是双$\mathrm{FP}_\infty$型，并且Hochschild上同调维数至多为$2$。 显示英文摘要

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi-$\mathrm{FP}_n$ if its group of units is of type $\mathrm{FP}_n$. We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular, we show that all such monoids are of type bi-$\mathrm{FP}_\infty$. Moreover, we show that if $r$ is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if $r$ is a proper power then it has infinite Hochschild cohomological dimension. For non-special one-relator monoids with defining relation $u=v$ we show that if there is no non-empty word $r$ such that $u,v \in A^*r \cap r A^*$ then $M$ is of type bi-$\mathrm{FP}_\infty$ and the Hochschild cohomological dimension at most $2$.

[10] arXiv:2507.05087 [中文pdf, pdf, html, 其他]

标题： 关于双曲群的子直接积的共轭问题 显示英文标题

标题： On the conjugacy problem for subdirect products of hyperbolic groups

Martin R. Bridson

评论： 17页，7图

主题： 群论 (math.GR)

如果$G_1$和$G_2$是无挠双曲群，且$P<G_1\times G_2$是一个有限生成的子直积，则$P$中的共轭问题可解当且仅当存在一个统一算法来判断有限表示群$G_1/(P\cap G_1)$中循环子群的成员资格。 该结果的证明依赖于一种新的技术，用于扰动双曲群中的元素，以确保它们不是真幂。 显示英文摘要

If $G_1$ and $G_2$ are torsion-free hyperbolic groups and $P<G_1\times G_2$ is a finitely generated subdirect product, then the conjugacy problem in $P$ is solvable if and only if there is a uniform algorithm to decide membership of the cyclic subgroups in the finitely presented group $G_1/(P\cap G_1)$. The proof of this result relies on a new technique for perturbing elements in a hyperbolic group to ensure that they are not proper powers.

交叉提交 (展示 5 之 5 条目 )

[11] arXiv:2507.03309 (交叉列表自 math.RT) [中文pdf, pdf, html, 其他]

标题： 无限对称群代数的一类特殊素理想 显示英文标题

标题： A special class of prime ideals for infinite symmetric group algebras

Kevin Coulembier

主题： 表示理论 (math.RT) ; 群论 (math.GR)

我们识别出有限无限对称群代数中一个有趣的素理想特殊类。 我们证明此类理想的集合携带一个半环结构。 在复数上，我们建立与（对应于）无限对称群的球面表示之间的联系。 在正特征情况下，我们研究其与对称张量范畴结构理论的紧密联系。 显示英文摘要

We identify an interesting special class of prime ideals in the finitary infinite symmetric group algebra. We show that the set of such ideals carries a semiring structure. Over the complex numbers, we establish a connection with spherical representations of (the Gelfand pair corresponding to) the infinite symmetric group. In positive characteristic, we investigate a close connection with the structure theory of symmetric tensor categories.

[12] arXiv:2507.03907 (交叉列表自 math.LO) [中文pdf, pdf, html, 其他]

标题： 关于同构关系的全源局部有限群 显示英文标题

标题： On the Isomorphism Relation for Omnigenous Locally Finite Groups

Su Gao, Feng Li

主题： 逻辑 (math.LO) ; 群论 (math.GR)

一个全包容局部有限群的概念在[2]中被引入，作为Hall的通用可数局部有限群的推广。 在本文中，我们证明了所有可数全包容局部有限群的类是Borel完全的，因此它在所有可数结构中具有最大的同构类型Borel基数。 [2] M. Etedadialiabadi, S. Gao, F. Le Maître, J. Melleray, 自动同构群中稠密的局部有限子群，Adv. Math. 391 (2021), 107966. 显示英文摘要

The concept of an omnigenous locally finite group was introduced in [2] as a generalization of Hall's universal countable locally finite group. In this paper we show that the class of all countable omnigenous locally finite groups is Borel complete, hence it has the maximum Borel cardinality of isomorphism types among all countable structures. [2] M. Etedadialiabadi, S. Gao, F. Le Ma\^{i}tre, J. Melleray, Dense locally finite subgroups of automorphism groups of ultraextensive spaces, Adv. Math. 391 (2021), 107966.

[13] arXiv:2507.04474 (交叉列表自 math.AG) [中文pdf, pdf, html, 其他]

标题： 有限子群的Cremona群表示 显示英文标题

标题： Representations of finite subgroups of Cremona groups

Alexander Duncan, Bailey Heath, Christian Urech

评论： 29页

主题： 代数几何 (math.AG) ; 群论 (math.GR)

克雷莫纳群在域k上的秩n是域k上n维射影空间的有理自同构的群。我们研究最小维度，使得克雷莫娜群的所有有限子群在相同域上有该维度的忠实表示。我们在所有域上找到了秩1和2的精确值。我们证明，在正特征域和秩大于1的情况下，该值是无限的。对于许多特征为0的域，包括数域和复数域，我们表明对于所有秩来说该值是有限的。最后，对于所有特征为0的域，我们证明该维度有一个下界，该下界随秩呈指数增长。 显示英文摘要

The Cremona group of rank n over a field k is the group of birational automorphisms of the n-dimensional projective space over the field k. We study the minimal dimension such that all finite subgroups of the Cremona group have a faithful representation of that dimension over the same field. We find the exact value for rank 1 and 2 over all fields. We prove that the value is infinite for all fields of positive characteristic and rank greater than one. For many fields of characteristic 0, which include number fields and the complex field, we show that the value is finite for all ranks. Finally, for all fields of characteristic 0, we prove that the dimension is bounded below by a function that is exponential in the rank.

[14] arXiv:2507.04691 (交叉列表自 math.OA) [中文pdf, pdf, html, 其他]

标题： II$_1$因子的 W* 相关性以及张量积和图积的刚性 显示英文标题

标题： W*-correlations of II$_1$ factors and rigidity of tensor products and graph products

Daniel Drimbe, Stefaan Vaes

主题： 算子代数 (math.OA) ; 群论 (math.GR)

群的Gromov测度等价概念的一个变体已被引入到II$_1$因子中，有不同的名称。 我们提出了W*-相关II$_1$因子的术语。 我们证明了在张量积和图积的II$_1$因子之间的W*-相关性下的刚性结果。 作为结果，我们构造了第一个不可数的离散群$\Gamma$家族，它们不是冯·诺依曼等价的，这意味着它们的群冯·诺依曼代数$L(\Gamma)$不是W*-相关的，并且这意味着这些群既不是测度等价的，也没有同构或几乎同构的群冯·诺依曼代数。 显示英文摘要

A variant of Gromov's notion of measure equivalence for groups has been introduced for II$_1$ factors under different names. We propose the terminology of W*-correlated II$_1$ factors. We prove rigidity results up to W*-correlations for tensor products and graph products of II$_1$ factors. As a consequence, we construct the first uncountable family of discrete groups $\Gamma$ that are not von Neumann equivalent, which means that their group von Neumann algebras $L(\Gamma)$ are not W*-correlated, and which implies that these groups are neither measure equivalent, nor have isomorphic or virtually isomorphic group von Neumann algebras.

[15] arXiv:2507.05033 (交叉列表自 math.DS) [中文pdf, pdf, html, 其他]

标题： 三次后临界有限多项式的有限几何迭代单色群 显示英文标题

标题： Profinite geometric iterated monodromy groups of postcritically finite polynomials in degree 3

Mikhail Hlushchanka, Olga Lukina, Dean Wardell

主题： 动力系统 (math.DS) ; 群论 (math.GR) ; 数论 (math.NT)

在本文中，我们研究了与多项式相关的有限几何迭代单色群的性质。 这些群可以看作是数域的绝对伽罗瓦群到正则根树自同构群的通用表示。 我们的主要结果是，对于一个数域上的3次后临界有限多项式，其中每个有限后临界点至少有一个不在临界轨道中的原像，所关联的有限几何迭代单色群是不可约有限生成的。 此外，该群由多项式的分歧图的同构类决定，最多通过三元根树的自同构共轭。 我们还研究了这类群的群论性质，即它们的分支和扭性质。 特别是，我们证明了这样的群在其换位子群闭包上是正则分支的，并且它们包含任何在三元树中可实现的阶数的扭元素。 显示英文摘要

In this article, we study the properties of profinite geometric iterated monodromy groups associated to polynomials. Such groups can be seen as generic representations of absolute Galois groups of number fields into the automorphism group of a regular rooted tree. Our main result is that, for a degree 3 postcritically finite polynomial over a number field, where each finite postcritical point has at least one preimage outside the critical orbits, the associated profinite geometric iterated monodromy group is finitely invariably generated. Moreover, this group is determined by the isomorphism class of the ramification portrait of the polynomial, up to conjugation by an automorphism of the ternary rooted tree. We also study the group-theoretical properties of such groups, namely their branch and torsion properties. In particular, we show that such groups are regular branch over the closure of their commutator subgroup, and that they contain torsion elements of any order realizable in the ternary tree.

替换提交 (展示 6 之 6 条目 )

[16] arXiv:2404.03952 (替换) [中文pdf, pdf, html, 其他]

标题： 置换群的最小生成集 显示英文标题

标题： Minimal sized generating sets of permutation groups

Derek F. Holt, Gareth Tracey

主题： 群论 (math.GR)

我们提出了一种Lucchini和Thakkar算法的随机变体，用于在有限群中找到最小大小的生成集，该算法在有限置换群中的期望运行时间是多项式时间。 显示英文摘要

We present a randomised variant of an algorithm of Lucchini and Thakkar for finding a smallest sized generating set in a finite group, which has polynomial time expected running time in finite permutation groups.

[17] arXiv:2408.15716 (替换) [中文pdf, pdf, html, 其他]

标题： 完全不连通局部紧致群的一些不变量：上同调与组合学 显示英文标题

标题： Some invariants of totally disconnected locally compact groups: cohomology and combinatorics

Ilaria Castellano, Bianca Marchionna, Thomas Weigel

评论： 31页。修正了拼写错误并重新排列了主题。相应地更改了标题、摘要和引言。

主题： 群论 (math.GR)

本文研究了完全不连通局部紧群的两个不变量：端点数和有理离散上同调维数。 对于这样的紧生成群$G$，证明了其端点数可以用$G$的有理离散上同调来表示。 如果$G$适当作用于一个建筑物上，那么$G$的端点数和有理上同调维数与该建筑物相关联的 Weyl 群的相应不变量有关。 在特殊情况下，我们还能够将$G$的有理离散上同调维数与$G$的平坦秩进行比较。 此外，给出了这两个不变量相等的群的例子。 我们的方法利用了 Coxeter 群的组合学，得到了在 Coxeter 理论中具有独立兴趣的新结果。 最后，在作用于局部有限建筑物上的完全不连通局部紧群类中，证明了一个可访问性结果：如果有理离散上同调维数为一，则我们显式构造了一个在树上的共紧且适当的行动。 显示英文摘要

The paper investigates two invariants for totally disconnected locally compact groups: the number of ends and the rational discrete cohomological dimension. For such a compactly generated group $G$ it is shown that its number of ends can be expressed in terms of the rational discrete cohomology of $G$. If $G$ is suitably acting on a building the number of ends and the rational cohomological dimension of $G$ are related to those of the Weyl group associated to the building. In special cases, we are also able to compare the rational discrete cohomological dimension of $G$ to the flat-rank of $G$. Moreover, examples of groups for which these two invariants coincide are given. Our approach leverages the combinatorics of Coxeter groups, yielding new results of independent interest in Coxeter theory. Finally, in the class of totally disconnected locally compact groups acting properly and cocompactly on locally finite buildings, an accessibility result is proved: we explicitly construct a cocompact proper action on a tree if the rational discrete cohomological dimension is one.

[18] arXiv:2504.12789 (替换) [中文pdf, pdf, html, 其他]

标题： 可解无平方因子群的枚举以及某些类型分裂扩张的计数 显示英文标题

标题： Enumeration of solvable cube-free groups and counting certain types of split extensions

Prashun Kumar, Geetha Venkataraman

评论： 传达的

主题： 群论 (math.GR)

如果一个群的阶不被任何素数的立方整除，则称该群为无立方因子群。 令$f_{cf,sol}(n)$表示阶为$n$的可解无立方因子群的同构类。 在本文中，我们找到了$f_{cf,sol}(n)$的渐近界。 设$p$为一个素数，令$q = p^k$对某个正整数$k$。 我们还给出了一个公式，用于计算在非交换可解无立方因子$p'$子群中最大的子群的共轭类数目，这些子群是${\rm GL}(2,q)$的子群。 此外，我们找到了在同构意义下，给定阶数的$P$对$Q$的分裂扩张的精确数目，其中$P \in \{{\mathbb Z}_p \times {\mathbb Z}_p, {\mathbb Z}_{p^{\alpha}}\}$，$p$是素数，$\alpha$是正整数，$Q$是奇数阶的无平方因子交换群，且$p \nmid |Q|$。 显示英文摘要

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this paper. Let $p$ be a prime and let $q = p^k$ for some positive integer $k$. We also give a formula for the number of conjugacy classes of the subgroups that are maximal amongst non-abelian solvable cube-free $p'$-subgroups of ${\rm GL}(2,q)$. Further, we find the exact number of split extensions of $P$ by $Q$ up to isomorphism of a given order where $P \in \{{\mathbb Z}_p \times {\mathbb Z}_p, {\mathbb Z}_{p^{\alpha}}\}$, $p$ is a prime, $\alpha$ is a positive integer and $Q$ is a cube-free abelian group of odd order such that $p \nmid |Q|$.

[19] arXiv:2507.02594 (替换) [中文pdf, pdf, html, 其他]

标题： 通过元素阶的乘积的素因数分解中的指数集合进行识别 显示英文标题

标题： Recognition by the set of exponents in the prime factorization of the product of element orders

Morteza Baniasad Azad, Mostafa Arabtash

主题： 群论 (math.GR)

设 $G$是一个有限群。 设 $\rho(G) = \prod_{g \in G} o(g)={p_1}^{\alpha_1} {p_2}^{\alpha_2} \cdots {p_k}^{\alpha_k}$，其中 $p_1, p_2, \cdots, p_k$是不同的素数，而 $o(g)$表示 $g \in G$的阶。 乘积元素阶的质因数分解中的指数集合记为 $ {\operatorname{Exp}}_{\rho}(G)$，即 $ {\operatorname{Exp}}_{\rho}(G)=\{\alpha_1,\alpha_2, \cdots,\alpha_k\}$。 在本文中，我们通过$ {\operatorname{Exp}}_{\rho}(G)$给出了一些群的新特征。 我们证明了群${\rm PSL}(2, 5) \times \mathbb{Z}_p$、${\rm PSL}(2, 7)$和${\rm PSL}(2, 11)$被$ {\operatorname{Exp}}_{\rho}(G)$唯一确定。 此外，我们证明了群 ${\rm PSL}(2, 5)$ 和 ${\rm PSL}(2, 13)$ 唯一地由参数 $ {\operatorname{Exp}}_{\rho}(G)$ 和 $|G|$ 确定。 此外，我们证明如果 ${\operatorname{Exp}}_{\rho}(G) = {\operatorname{Exp}}_{\rho}(\mathbb{Z}_{2qr})$，则 $G \cong {\rm PSL}(2, 5)$ 或 $G \cong \mathbb{Z}_{2qr}$，其中 $q$ 和 $r$ 是不同的奇素数。 显示英文摘要

Let $G$ be a finite group. Let $\rho(G) = \prod_{g \in G} o(g)={p_1}^{\alpha_1} {p_2}^{\alpha_2} \cdots {p_k}^{\alpha_k}$, where $p_1, p_2, \cdots, p_k$ are distinct prime numbers and $o(g)$ denotes the order of $g \in G$. The set of exponents in the prime factorization of the product of element orders is denoted by $ {\operatorname{Exp}}_{\rho}(G)$, i.e., $ {\operatorname{Exp}}_{\rho}(G)=\{\alpha_1,\alpha_2, \cdots,\alpha_k\}$. In this paper, we give a new characterization for some groups by $ {\operatorname{Exp}}_{\rho}(G)$. We prove that the groups ${\rm PSL}(2, 5) \times \mathbb{Z}_p$, ${\rm PSL}(2, 7)$ and ${\rm PSL}(2, 11)$ are uniquely determined by $ {\operatorname{Exp}}_{\rho}(G)$. Furthermore, we prove that the groups ${\rm PSL}(2, 5)$ and ${\rm PSL}(2, 13)$ are uniquely determined by the parameters $ {\operatorname{Exp}}_{\rho}(G)$ and $|G|$. Additionally, we prove that if ${\operatorname{Exp}}_{\rho}(G) = {\operatorname{Exp}}_{\rho}(\mathbb{Z}_{2qr})$, then $G \cong {\rm PSL}(2, 5)$ or $G \cong \mathbb{Z}_{2qr}$, where $q$ and $r$ are distinct odd prime numbers.

[20] arXiv:2504.06893 (替换) [中文pdf, pdf, html, 其他]

标题： 局部域的仿射群是厄米特的 显示英文标题

标题： The affine group of a local field is Hermitian

Max Carter

评论： 6页。与前一版本相比有少量阐述修改。将发表于《Archiv der Mathematik》

主题： 泛函分析 (math.FA) ; 群论 (math.GR)

该群$\mathbb{Q}_p \rtimes \mathbb{Q}_p^*$是否为埃尔米特群的问题在文献中的多个来源中被列为开放问题，甚至最近在R. Palma于2015年发表的一篇论文中也提到了这个问题。在本文中，我们通过证明以下更一般的定理来确认该群是埃尔米特群：给定任何局部域$\mathbb{K}$，仿射群$\mathbb{K} \rtimes \mathbb{K}^*$是一个埃尔米特群。该证明是1970年代关于埃尔米特巴拿赫$*$-代数的结果的结论。在$\mathbb{K}$是非阿基米德局部域的情况下，这个结果产生了具有指数增长的完全不连通局部紧致埃尔米特群的例子，这些是满足这些性质的群的第一个例子。这回答了Palma关于此类群存在性的第二个问题。 显示英文摘要

The question of whether the group $\mathbb{Q}_p \rtimes \mathbb{Q}_p^*$ is Hermitian has been stated as an open question in multiple sources in the literature, even as recently as a paper by R. Palma published in 2015. In this note we confirm that this group is Hermitian by proving the following more general theorem: given any local field $\mathbb{K}$, the affine group $\mathbb{K} \rtimes \mathbb{K}^*$ is a Hermitian group. The proof is a consequence of results about Hermitian Banach $*$-algebras from the 1970's. In the case that $\mathbb{K}$ is a non-archimedean local field, this result produces examples of totally disconnected locally compact Hermitian groups with exponential growth, and these are the first examples of groups satisfying these properties. This answers a second question of Palma about the existence of such groups.

[21] arXiv:2506.04882 (替换) [中文pdf, pdf, html, 其他]

标题： 等周不等式在渐近秩为二的Hadamard空间中 显示英文标题

标题： Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

Urs Lang, Stephan Stadler, David Urech

评论： 21页，小修改

主题： 度量几何 (math.MG) ; 微分几何 (math.DG) ; 群论 (math.GR)

格罗莫夫关于哈达玛空间的等周间隙猜想指出，维度大于或等于渐近秩的循环具有线性等周填充不等式，而非低维的欧几里得型不等式。 在渐近秩为2的情况下，德鲁图-朗-帕帕索格卢-斯塔德勒最近取得了进展，他们建立了指数接近1的利普希茨2球面的同伦不等式。 我们证明了在环境空间具有有限线性控制渐近维数的假设下，至少二维的一般循环具有相同类型的同调不等式。 这尤其适用于所有哈达玛3流形和有限维CAT(0)立方复形。 显示英文摘要

Gromov's isoperimetric gap conjecture for Hadamard spaces states that cycles in dimensions greater than or equal to the asymptotic rank admit linear isoperimetric filling inequalities, as opposed to the inequalities of Euclidean type in lower dimensions. In the case of asymptotic rank 2, recent progress was made by Dru\c{t}u-Lang-Papasoglu-Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. We prove a homological inequality of the same type for general cycles in dimensions at least 2, assuming that the ambient space has finite linearly controlled asymptotic dimension. This holds in particular for all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.