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[1] arXiv:2508.02859 [中文pdf, pdf, html, 其他]

标题： 线性群的$2$维度 显示英文标题

标题： The essential $2$-dimension of the linear groups

Hannah Knight

评论： arXiv管理员注释：与arXiv:2109.02698和arXiv:2204.13227文本重叠

主题： 群论 (math.GR) ; 代数几何 (math.AG) ; 表示理论 (math.RT)

在本文中，我们计算了当定义素数为奇数时一般线性群、射影一般线性群、当$n$为奇数或$n = 2$时的特殊线性群，以及在$q \equiv 1 \mod 4$、$s = v_2(q-1)$和$\Gamma = \text{Gal}(k(\zeta_{2^s})/k)$为平凡的情况下特殊线性群及其商群（如射影特殊线性群）的本质$2$维度。 显示英文摘要

In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n = 2$, as well as the special linear groups and quotients of it (such as the projective special linear groups) in the case case $q \equiv 1 \mod 4$, $s = v_2(q-1)$, and $\Gamma = \text{Gal}(k(\zeta_{2^s})/k)$ is trivial.

[2] arXiv:2508.03056 [中文pdf, pdf, html, 其他]

标题： 对称群$S_n, n\geq 4$和有限非交换单群不能嵌入任何Riordan群中 显示英文标题

标题： The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group

Tian-Xiao He, Nikolai A. Krylov

评论： 19页

主题： 群论 (math.GR) ; 组合数学 (math.CO)

我们证明，阶大于三的对称群无法嵌入到系数在任意交换环上的Riordan群中。 我们还证明了无法嵌入有限非阿贝尔单群。 作为紧密相关的话题，我们说明为什么所有截断的Riordan群都是可解的，这与无限大小的Riordan群的不可解性形成鲜明对比。 最后，我们给出了交替群$A_4$在某个交换环系数的Lagrange子群中的显式嵌入，并证明$A_4$无法嵌入到一个替换群中，因此也无法嵌入到一个Nottingham群中。 显示英文摘要

We prove that the symmetric group of degree greater than three cannot be embedded into the Riordan group with coefficients in any commutative ring. We also prove the impossibility to embed finite non-abelian simple groups. As a closely related topic, we show why all truncated Riordan groups are solvable, in stark contrast to the unsolvability of the infinite-sized Riordan groups. Finally, we give an explicit embedding of the alternating group $A_4$ into the Lagrange subgroup with coefficients in a certain commutative ring, and prove that $A_4$ cannot be embedded into a substitution group, and hence, a Nottingham group.

[3] arXiv:2508.03231 [中文pdf, pdf, html, 其他]

标题： 关于广义Baumslag-Solitar群的同构问题：角度 显示英文标题

标题： On the isomorphism problem for generalized Baumslag-Solitar groups: angles

Dario Ascari, Montserrat Casals-Ruiz, Ilya Kazachkov

评论： 31页，17图

主题： 群论 (math.GR)

我们引入了一种新的广义Baumslag-Solitar (GBS)群的同构不变量，我们称之为极限角。 与之前已知的主要是代数的不变量不同，极限角具有动态解释，仅在两个相互作用的边的情况下出现。 这个不变量捕捉到了在具有更多相互作用边的配置中不会显现的细微几何行为。 作为应用，我们使用极限角对只有一个顶点和两条边的GBS群进行分类。 显示英文摘要

We introduce a new isomorphism invariant for generalized Baumslag-Solitar (GBS) groups, which we call the limit angle. Unlike previously known invariants, which are primarily algebraic, the limit angle admits a dynamical interpretation, arising exclusively in the case of two interacting edges. This invariant captures subtle geometric behavior that does not manifest in configurations with more interacting edges. As an application, we use the limit angle to obtain a classification of GBS groups with one vertex and two edges.

[4] arXiv:2508.03372 [中文pdf, pdf, html, 其他]

标题： 一种计算低次数Hopf--Galois结构和斜括号的算法 显示英文标题

标题： An Algorithm for Computing Hopf--Galois Structures and Skew Bracoids of Low Degree

Andrew Darlington

评论： 23页。欢迎提出评论和建议！

主题： 群论 (math.GR)

利用分离扩张上的Hopf-Galois结构和斜括号系都与全纯的传递子群有内在联系这一事实，我们提出一种算法来对这些对象在几个低次数情况下进行分类和枚举。 我们的方法使我们能够大大扩展现有的关于Hopf-Galois结构的计算结果，同时建立在该领域中已有的算法之上。 显示英文摘要

Making use of the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph, we present an algorithm to classify and enumerate these objects in several low degree cases. Our approach allows us to greatly extend existing computational results for Hopf-Galois structures, while building on the existing algorithms presented in this area.

[5] arXiv:2508.03384 [中文pdf, pdf, html, 其他]

标题： 与Cunningham链相关的可分域扩张的Hopf-Galois结构 显示英文标题

标题： Hopf-Galois structures on separable field extensions of degree related to Cunningham chains

Andrew Darlington

评论： 27页。欢迎提出评论和建议！

主题： 群论 (math.GR)

过去几年中，人们在各种背景下研究了平方自由次数扩张上的Hopf--Galois结构。Galois情形由Alabdali和Byott在2020年完全探讨，随后Byott和Martin-Lyons首次尝试将这些结果推广到包括非正规扩张的情况；他们的工作研究了次数为$pq$的可分扩张，其中包含$p,q$个不同的奇素数，以及$p=2q+1$。本文进一步扩展了后者的成果，考虑了平方自由次数为$n=p_1...p_m$的可分扩张，其中每对连续的素数$p_i,p_{i+1}$由$p_i=2p_{i+1}+1$相关联。 显示英文摘要

The past few years have seen Hopf--Galois structures on extensions of squarefree degree studied in various contexts. The Galois case was fully explored by Alabdali and Byott in 2020, followed by a first attempt at generalising these results to include non-normal extensions by Byott and Martin-Lyons; their work looks at separable extensions of degree $pq$ with $p,q$ distinct odd primes, and $p=2q+1$. This paper extends the latter work further by considering separable extensions of squarefree degree $n=p_1...p_m$ where each pair of consecutive primes $p_i,p_{i+1}$ are related by $p_i=2p_{i+1}+1$.

[6] arXiv:2508.03479 [中文pdf, pdf, html, 其他]

标题： 关于单群中幂零子群的交集 显示英文标题

标题： On the intersections of nilpotent subgroups in simple groups

Timothy C. Burness, Hong Yi Huang

评论： 37页

主题： 群论 (math.GR)

设$G$是一个有限群，设$H_p$是$G$的一个西罗$p$-子群。 一个最近的由Lisi和Sabatini提出的猜想断言存在一个元素$x \in G$，使得$H_p \cap H_p^x$在集合$\{H_p \cap H_p^g \,:\, g \in G\}$中对于每个素数$p$都是包含最小的。这已在几个特殊情况下得到证明，包括所有足够大的对称群和交替群。 对于一个单群$G$，根据Mazurov和Zenkov于1996年的一个定理，该猜想意味着存在一个元素$x \in G$，使得对于所有$p$都有$H_p \cap H_p^x = 1$。 与此相一致，这一陈述暗示了Vdovin在2002年提出的猜想，该猜想断言如果$G$是单的且$H$是幂零子群，则$H \cap H^x = 1$对某些$x \in G$成立。 在本文中，我们采用概率方法证明了所有非交替单群的Lisi-Sabatini猜想。 通过结合Kurmazov关于交替群的幂零子群的早期工作，我们完成了对Vdovin猜想的证明。 此外，通过将我们对李型群的证明与Zenkov关于交替群和散在群的早期工作相结合，我们能够建立Vdovin猜想的一个更强形式：如果$G$是单群且$A,B$是幂零子群，则$A \cap B^x = 1$对于某个$x \in G$成立。 此外，我们研究了在李型单群中随机一对Sylow$p$-子群相交于平凡子群的渐近概率，补充了Diaconis等人和Eberhard关于对称群和交替群的近期工作。 显示英文摘要

Let $G$ be a finite group and let $H_p$ be a Sylow $p$-subgroup of $G$. A very recent conjecture of Lisi and Sabatini asserts the existence of an element $x \in G$ such that $H_p \cap H_p^x$ is inclusion-minimal in the set $\{H_p \cap H_p^g \,:\, g \in G\}$ for every prime $p$. This has been proved in several special cases, including all sufficiently large symmetric and alternating groups. For a simple group $G$, in view of a theorem of Mazurov and Zenkov from 1996, the conjecture implies the existence of an element $x \in G$ with $H_p \cap H_p^x = 1$ for all $p$. In turn, this statement implies a conjecture of Vdovin from 2002, which asserts that if $G$ is simple and $H$ is a nilpotent subgroup, then $H \cap H^x = 1$ for some $x \in G$. In this paper, we adopt a probabilistic approach to prove the Lisi-Sabatini conjecture for all non-alternating simple groups. By combining this with earlier work of Kurmazov on nilpotent subgroups of alternating groups, we complete the proof of Vdovin's conjecture. Moreover, by combining our proof for groups of Lie type with earlier work of Zenkov on alternating and sporadic groups, we are able to establish a stronger form of Vdovin's conjecture: if $G$ is simple and $A,B$ are nilpotent subgroups, then $A \cap B^x = 1$ for some $x \in G$. In addition, we study the asymptotic probability that a random pair of Sylow $p$-subgroups in a simple group of Lie type intersect trivially, complementing recent work of Diaconis et al. and Eberhard on symmetric and alternating groups.

[7] arXiv:2508.03632 [中文pdf, pdf, html, 其他]

标题： 关于逆半群的零因子图的直径和围长 显示英文标题

标题： On the diameter and girth of zero-divisor graphs of inverse semigroups

Yanhui Wang, Xinyi Zhu, Pei Gao

评论： 14页

主题： 群论 (math.GR) ; 组合数学 (math.CO)

设$S$是一个含零元的逆半群，令$Z(S)^\times$是其关于自然偏序$\le $在$S$上的非零因子集合，即，若存在$b\in S\setminus\{0\}$使得$\omega(a, b) = \{c \in S: c \leq a\ \mbox{and}\ c \leq b\}=\{0\}$，则$a \in Z(S)^\times $。 集合$Z(S)^\times$构成了相应{\it 零因子图} $\Gamma (S)$ 的顶点，若$\omega(a, b)=\{0\}$，则两个不同的顶点$a, b$形成一条边。 我们通过其直径和围长来表征逆半群的{\it 零因子图}。 我们还通过建立无零元的逆半群的直径（围长）与最小群同余$\sigma$之间的联系，对无零元的逆半群进行分类。 最后，我们根据图 $G$的顶点集和边集来描述图逆半群 $I(G)$的直径和围长。 显示英文摘要

Let $S$ be an inverse semigroup with zero and let $Z(S)^\times$ be its set of non-zero divisors with respect to the natural partial order $\le $ on $S$, that is, $a \in Z(S)^\times $ if there exists $b\in S\setminus\{0\}$ with $\omega(a, b) = \{c \in S: c \leq a\ \mbox{and}\ c \leq b\}=\{0\}$. The set $Z(S)^\times$ makes up the vertices of the corresponding {\it zero-divisor graph} $\Gamma (S)$, with two distinct vertices $a, b$ forming an edge if $\omega(a, b)=\{0\}$. We characterize {\it zero-divisor graphs} of inverse semigroups in terms of their diameter and girth. We also classify inverse semigroups without zero by building a connection between the diameter (girth) and the least group congruence $\sigma$ on an inverse semigroup without zero. Finally, we give a description of the diameter and girth of graph inverse semigoups $I(G)$ in terms of the set of vertices and the set of edges of a graph $G$.

[8] arXiv:2508.03648 [中文pdf, pdf, html, 其他]

标题： 有限群，其中每个真特征子群都是循环的 显示英文标题

标题： Finite groups in which every proper characteristic subgroup is cyclic

Marco Damele, Fabio Mastrogiacomo

评论： 17页

主题： 群论 (math.GR)

设$G$是一个有限的非循环、非特征简单的群，具有所有$G$的真特征子群都是循环的性质。 我们称这样的群为$\mathrm{CCS}$群，简称\emph{特征循环}。 在本文中，我们提供了这些群的完整分类。 作为我们主要结果的应用，我们也朝着对最小非循环斜环的分类取得了一些进展。 显示英文摘要

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{CCS}$ group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.

交叉提交 (展示 1 之 1 条目 )

[9] arXiv:2508.03662 (交叉列表自 math.OA) [中文pdf, pdf, html, 其他]

标题： 图积冯诺依曼代数的刚性 显示英文标题

标题： Rigidity for graph product von Neumann algebras

Camille Horbez, Adrian Ioana

主题： 算子代数 (math.OA) ; 群论 (math.GR)

我们建立了与有限简单图$\Gamma$相关的图乘积冯诺依曼代数$M_\Gamma=*_{v,\Gamma}M_v$和迹冯诺依曼代数族$(M_v)_{v\in\Gamma}$的刚性定理。 我们考虑以下三类顶点代数：扩散型、扩散可约型和II$_1$因子。 在这些三个区域中的每一个中，我们展示了一个大的图类$\Gamma,\Lambda$，其中以下情况成立：任何同构$\theta$在$M_\Gamma$和$N_\Lambda$之间确保存在一个图同构$\alpha:\Gamma\to\Lambda$，并且对于每个顶点$v\in\Gamma$，$\theta(M_v)$和$N_{\alpha(v)}$之间有紧密的关系，从双方的强交织（根据 Popa 的意义），到某些情况下的酉共轭。 我们的结果导致了对图积冯诺依曼代数分类以及计算其对称群的广泛应用。 首先，我们获得了右角Artin群的冯诺依曼代数和ICC群的图积的一般分类定理。 我们还提供了一类具有平凡基本群的II$_1$因子，包括所有在至少具有$5$圈长的图上以及没有度数为$0$或$1$的顶点的II$_1$因子的图积。 最后，我们计算了某些II$_1$因子的图积的外自同构群。 显示英文摘要

We establish rigidity theorems for graph product von Neumann algebras $M_\Gamma=*_{v,\Gamma}M_v$ associated to finite simple graphs $\Gamma$ and families of tracial von Neumann algebras $(M_v)_{v\in\Gamma}$. We consider the following three broad classes of vertex algebras: diffuse, diffuse amenable, and II$_1$ factors. In each of these three regimes, we exhibit a large class of graphs $\Gamma,\Lambda$ for which the following holds: any isomorphism $\theta$ between $M_\Gamma$ and $N_\Lambda$ ensures the existence of a graph isomorphism $\alpha:\Gamma\to\Lambda$, and tight relations between $\theta(M_v)$ and $N_{\alpha(v)}$ for every vertex $v\in\Gamma$, ranging from strong intertwining in both directions (in the sense of Popa), to unitary conjugacy in some cases. Our results lead to a wide range of applications to the classification of graph product von Neumann algebras and the calculation of their symmetry groups. First, we obtain general classification theorems for von Neumann algebras of right-angled Artin groups and of graph products of ICC groups. We also provide a new family of II$_1$ factors with trivial fundamental group, including all graph products of II$_1$ factors over graphs with girth at least $5$ and no vertices of degree $0$ or $1$. Finally, we compute the outer automorphism group of certain graph products of II$_1$ factors.

替换提交 (展示 8 之 8 条目 )

[10] arXiv:2305.14545 (替换) [中文pdf, pdf, html, 其他]

标题： 群的Liouville性质和共形维数 显示英文标题

标题： Liouville property for groups and conformal dimension

Nicolás Matte Bon, Volodymyr Nekrashevych, Tianyi Zheng

评论： v4：42页，5张图，最终版本

主题： 群论 (math.GR) ; 动力系统 (math.DS) ; 概率 (math.PR)

共形维数是度量空间的基本不变量，特别适合于自相似空间的研究，例如具有扩张自覆盖的空间（例如复有理函数的Julia集）。这些系统的动力学由相关的迭代单值群编码，这些是收缩自相似群的例子。它们的可约性是一个众所周知的开放问题。我们证明，如果$G$是一个迭代单值群，并且底层空间的（Alfhors-正则）共形维数严格小于2，那么在$G$上的每个具有有限二阶矩的对称随机游走都具有Liouville性质。作为推论，每个这样的群都是可约的。此准则适用于所有之前已知的可约收缩群例子以及许多新的例子。特别是，它意味着对于每一个子双曲复有理函数$f$，其Julia集不是整个球面，那么$f$的迭代单值群是可约的。 显示英文摘要

Conformal dimension is a fundamental invariant of metric spaces, particularly suited to the study of self-similar spaces, such as spaces with an expanding self-covering (e.g. Julia sets of complex rational functions). The dynamics of these systems are encoded by the associated iterated monodromy groups, which are examples of contracting self-similar groups. Their amenability is a well-known open question. We show that if $G$ is an iterated monodromy group, and if the (Alfhors-regular) conformal dimension of the underlying space is strictly less than 2, then every symmetric random walk with finite second moment on $G$ has the Liouville property. As a corollary, every such group is amenable. This criterion applies to all examples of contracting groups previously known to be amenable, and to many new ones. In particular, it implies that for every sub-hyperbolic complex rational function $f$ whose Julia set is not the whole sphere, the iterated monodromy group of $f$ is amenable.

[11] arXiv:2503.01366 (替换) [中文pdf, pdf, html, 其他]

标题： 斜括号中下中央序列和上中央序列的类似物：综述 显示英文标题

标题： Analogs of the lower and upper central series in skew braces: a survey

Cindy Tsang

评论： 30页；最终版本将被发表

主题： 群论 (math.GR) ; 量子代数 (math.QA)

一个斜环是一种类似环和群的代数结构，在研究Yang-Baxter方程的集合论解时被引入。 在本文综述中，我们将考虑斜环的左序列、右序列、幂零序列和消去序列。 它们可以看作是群的下中心序列和上中心序列的类似物。 除了这些序列的一些已知事实外，我们将证明关于它们项之间关系的几个新结果。 我们还将考虑由Bonatto和Jedlička定义的斜环的下中心序列。 正如我们将解释的那样，它似乎是对斜环的下中心序列的“正确”类比。 关于这一点，我们还将讨论Ballester-Bolinches等人提出的理想下中心序列的概念。 显示英文摘要

A skew brace is a ring-like and group-like algebraic structure that was introduced in the study of set-theoretic solutions to the Yang-Baxter equation. In this survey paper, we shall consider the left series, right series, socle series, and annihilator series of skew braces. They may be regarded as analogs of the lower and upper central series of groups. Other than some well-known facts regarding these series, we shall prove several new results about the relationship among their terms. We shall also consider the lower central series of skew braces that was defined by Bonatto and Jedli\v{c}ka. As we shall explain, it seems to be the ``correct" analog of the lower central series for skew braces. Concerning this, we shall also discuss the notion of the lower central series of ideals that is due to Ballester-Bolinches et al.

[12] arXiv:2506.19745 (替换) [中文pdf, pdf, html, 其他]

标题： 几乎单群中Sylow子群的交集 显示英文标题

标题： On the intersections of Sylow subgroups in almost simple groups

Timothy C. Burness, Hong Yi Huang

评论： 26页；小幅度调整

主题： 群论 (math.GR)

设$G$是一个有限的几乎单群，令$H$是$G$的一个西罗$p$-子群。 作为Zenkov定理的一个特例，存在$x,y \in G$使得$H \cap H^x \cap H^y = 1$。 事实上，如果$G$是单的，那么 Mazurov 和 Zenkov 的一个定理表明，$H \cap H^x = 1$对某些$x \in G$成立。然而，已知该性质并不适用于所有几乎单群。例如，如果$G = S_8$和$p=2$，则$H \cap H^x \ne 1$对所有$x \in G$成立。 Zenkov在1990年代的进一步工作表明，这样的例子很少（例如，如果$p \geqslant 5$，则没有这样的例子），他将所有这些对的分类简化为$p=2$和$G$是一个定义在有限域$\mathbb{F}_q$上的几乎单李型群的情况，并且$q=9$或$q$是一个梅森素数或费马素数。在本文中，通过采用基于固定点比率估计的概率方法，我们完成了Zenkov的分类。 显示英文摘要

Let $G$ be a finite almost simple group and let $H$ be a Sylow $p$-subgroup of $G$. As a special case of a theorem of Zenkov, there exist $x,y \in G$ such that $H \cap H^x \cap H^y = 1$. In fact, if $G$ is simple, then a theorem of Mazurov and Zenkov reveals that $H \cap H^x = 1$ for some $x \in G$. However, it is known that the latter property does not extend to all almost simple groups. For example, if $G = S_8$ and $p=2$, then $H \cap H^x \ne 1$ for all $x \in G$. Further work of Zenkov in the 1990s shows that such examples are rare (for instance, there are no such examples if $p \geqslant 5$) and he reduced the classification of all such pairs to the situation where $p=2$ and $G$ is an almost simple group of Lie type defined over a finite field $\mathbb{F}_q$ and either $q=9$ or $q$ is a Mersenne or Fermat prime. In this paper, by adopting a probabilistic approach based on fixed point ratio estimates, we complete Zenkov's classification.

[13] arXiv:2508.01949 (替换) [中文pdf, pdf, html, 其他]

标题： 将丰富的半群作为逆半群的(2,1,1)-子代数嵌入 显示英文标题

标题： Embedding ample semigroups as (2,1,1)-subalgebras of inverse semigroups

Nasir Sohail, Aftab Hussain Shah, Kristo Väljako

评论： 无

主题： 群论 (math.GR)

嵌入一个充分半群作为(2,1,1)型子代数到一个逆半群中的问题是已知不可判定的。 在本文中，我们研究了某些类别的充分半群的这个问题。 我们还给出了左（分别右）但不是右（分别左）充分的半群的例子。 显示英文摘要

The problem of embedding an ample semigroup in an inverse semigroup as a (2, 1, 1)-type subalgebra is known to be undecidable. In this article, we investigate the problem for certain classes of ample semigroups. We also give examples of semigroups that are left (respectively, right) but not right (respectively, left) ample.

[14] arXiv:2003.11212 (替换) [中文pdf, pdf, html, 其他]

标题： 曲面映射类群上的扭共有限拓扑 显示英文标题

标题： The twist-cofinite topology on the mapping class group of a surface

Ingrid Irmer

主题： 几何拓扑 (math.GT) ; 群论 (math.GR)

在紧致连通可定向曲面上的映射类群上定义了一个拓扑。 从这个定义中得出了一种“普遍性”在映射类群子集上的概念。 从这个概念很容易得出许多合理的结论；例如，伪Anosov映射的集合被证明是普遍的，并且可以假设其具有任意大的拉伸因子，普遍地。 设M是从固定亏格g的Heegaard分解和普遍粘合映射得到的3-流形。 对于这样的流形，普遍地M是双曲的，第一Betti数为零，并且Heegaard亏格恰好等于g。 显示英文摘要

A topology is defined on the mapping class group of a compact connected orientable surface. It is shown that a notion of "genericity" on subsets of the mapping class group arises from this definition. Many plausible results follow from this notion easily; for example, the set of pseudo-Anosov maps is shown to be generic, and can be assumed to have arbitrary large stretch factor, generically. Let M be a 3-manifold obtained from a Heegaard splitting of fixed genus g and generic gluing map. It is shown that for such manifolds, generically M is hyperbolic, has first Betti number zero and Heegaard genus exactly equal to g.

[15] arXiv:2211.03429 (替换) [中文pdf, pdf, html, 其他]

标题： Thurston脊的等变形变收缩 显示英文标题

标题： An equivariant deformation retraction of the Thurston spine

Ingrid Irmer

评论： 这篇论文被分为三部分。对Thurston工作的综述在这里“Thurston的Teichmuller空间变形收缩”，即将发表在“Thurston传统第四部”中，第二部分是“填充曲线的拟阵性质”，即将发表在“澳大利亚数学学会公报”中。此版本对应原论文的最后一章。

主题： 几何拓扑 (math.GT) ; 群论 (math.GR)

本文表明，闭合可定向曲面的Teichmüller空间可以通过映射类群等变的形变收缩映射到一个维数等于映射类群的虚拟上同调维数的单元复形。 形变收缩映射的像是包含在Thurston首次描述的CW复形中——即Thurston脊。 Thurston脊是Teichmüller空间中对应于双曲曲面的点的集合，这些双曲曲面的最短测地线（即系统线）将曲面切割成多边形。 显示英文摘要

This paper shows that there is a mapping class group-equivariant deformation retraction of the Teichm\"uller space of a closed, orientable surface onto a cell complex of dimension equal to the virtual cohomological dimension of the mapping class group. The image of the deformation retraction is contained in the CW complex first described by Thurston -- the Thurston spine. The Thurston spine is the set of points in Teichm\"uller space corresponding to hyperbolic surfaces for which the set of shortest geodesics (the systoles) cuts the surface into polygons.

[16] arXiv:2401.15306 (替换) [中文pdf, pdf, html, 其他]

标题： 关于$2$-积分Cayley图 显示英文标题

标题： On $2$-integral Cayley graphs

Alireza Abdollahi, Majid Arezoomand, Tao Feng, Shixin Wang

主题： 组合数学 (math.CO) ; 群论 (math.GR)

在本文中，我们引入了$k$-积分图的概念。一个图$\Gamma$被称为$k$-积分，如果特征多项式$\Gamma$的分裂域在有理数域$\mathbb Q$上的扩张次数等于$k$。我们证明了所有具有给定代数次数和最大次数的有限连通图的集合是有限的。$1$-积分图就是积分图，即所有特征值都是整数的图。 我们研究有限群$2$上的$G$-积分Cayley图，相对于Cayley集，这些集合是$G$的共轭类的并集。在其他一般结果中，我们完全表征了所有具有连通$2$-积分Cayley图且度数为$2,3,4$和$5$的有限阿贝尔群。 此外，我们分类有限群$G$，其中所有在$G$上的有界度数的Cayley图都是$2$-整数。 显示英文摘要

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to $k$. We prove that the set of all finite connected graphs with given algebraic degree and maximum degree is finite. $1$-integral graphs are just integral ones, graphs all of whose eigenvalues are integer. We study $2$-integral Cayley graphs over finite groups $G$ with respect to Cayley sets which are a union of conjugacy classes of $G$. Among other general results, we completely characterize all finite abelian groups having a connected $2$-integral Cayley graph with valency $2,3,4$ and $5$. Furthermore, we classify finite groups $G$ for which all Cayley graphs over $G$ with bounded valency are $2$-integral.

[17] arXiv:2508.01347 (替换) [中文pdf, pdf, 其他]

标题： 廉价嵌入原理：同调增长的动力学上界 显示英文标题

标题： The cheap embedding principle: Dynamical upper bounds for homology growth

Kevin Li, Clara Loeh, Marco Moraschini, Roman Sauer, Matthias Uschold

评论： 104页，标题已更正

主题： 代数拓扑 (math.AT) ; 动力系统 (math.DS) ; 群论 (math.GR) ; 算子代数 (math.OA)

我们提供了群的对数扭同调增长和贝蒂数增长的上界，用测度群理论的语言来表述。 显示英文摘要

We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.