Generalised triangle groups of type (3,3,2)
James Howie
如果G是一个表示形式为< x，y的群|x^3=y^3=W（x，y）^2=1 >，则G是虚可解的或G包含秩为2的自由子群。这为罗森伯格的一个猜想提供了额外的证据。
If G is a group with a presentation of the form < x,y|x^3=y^3=W(x,y)^2=1 >, then either G is virtually soluble or G contains a free subgroup of rank 2. This provides additional evidence in favour of a conjecture of Rosenberger.
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Generalised triangle groups of type (2,3,2) with no cyclic essential representations
本研究针对Roseberger关于广义三角群的猜想，在两种例外情形下验证了无非循环本质表示的(2,3,2)型广义三角群满足该猜想。
Generalised triangle groups of type (3,5,2)
如果G是一个表示形式为< x，y的群|x^3=y^5=W（x，y）^2=1 >，则G是虚可解的或G含有秩为2的自由子群。这为罗森伯格的一个猜想提供了额外的证据。
On One-relator products induced by generalised triangle groups
本文研究了由广义三角群诱导的单关系子积群，利用广义三角群性质及对图像和自由幺半群中字的分析，在较弱假设下证明了 Freiheitssatz 和 Mayer-Vietoris 序列的存在性，推广了前人结果。
On Thin Heckoid and Generalised Triangle Groups in $PSL(2,\mathbb{C})$}
该研究识别 $PSL(2,\mathbb{C})$ 中的薄 Heckoid 群，并完整列出斜率为 $1/2$ 的 55 个薄广义三角群。
The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$
排序局部有限偏序集$P$的$M$ -三角形是生成函数$\sum_{u,w\in P} ^{}μ(u,w) x^{\rk u}y^{\rk w}$，其中$μ(.,.)$是$P$的莫比乌斯函数。本文计算了$E_7$和$E_8$型根系的$m$ -可除非交叉划分的Armstrong偏序集的$M$ -三角形。对于除$D_n$之外的其他类型，这已经在早期的论文“广义簇复合物的$F$ -三角形”中完成。“总之，这几乎解决了阿姆斯特朗的$F=M$猜想，该猜想预测了$m$ -可分分区偏序集的$M$ -三角形与Fomin和Reading的广义簇复合体的$F$ -三角形（某种精确的面数）之间令人惊讶的关系，这是$D_n$类型中唯一剩下的差距。此外，我们证明了一个互惠的结果，这个$M$ -三角形，再次与类型$D_n$可能的例外。我们的研究结果是基于计算的某些分解数的反射群类型$E_7$和$E_8$，实际上进行更精细的信息比$M$ -三角形。其他特殊反射群的分解数在前面的论文中已经计算过了。给出了$A_n$型分解数的一个简化公式.
Generalised Triangle Groups of Type (2,4,2)
本文研究Rosenberger猜想在(2,4,2)情形下的进展，证明反例中词W的指数和奇偶性限制及自由积长度至少68，计算机调查给出更强下界196。
Resolutions of orbifold generalised triangle groups
本文研究orbifold广义三角群及其中心扩张的自由分解，计算了相关群的整数同调上同调，对比了不同有理斜率族的分解秩差异。
The generalised Fitting subgroup of a profinite group
有限群的广义Fitting子群是由所有幂零或拟单的次正规子群生成的群。这个子群在有限群理论中的重要性源于这样一个事实，即它总是包含自己的中心化器，因此任何有限群都是其广义Fitting子群的自同构自同构群的阿贝尔扩展。我们定义了一类推广这一现象的投射有限群，并探讨了投射有限群的结构的一些后果。
Fixed points of finite groups acting on generalised Thompson groups
本文研究了Thompson群F的推广的有穷级自同构的中心化子和这些群的有限扩张中有限子群的共轭类。特别是，我们表明，在这些广义汤普森群的有限自同构的中心化要么承认一个有限型分类空间或不生成的。作为应用，我们推出了这类有限扩张的Bredon型结果.
Generation of Generalised Wreath Products of Symmetric Groups
研究对称群序列的广义 wreath 乘积生成，确定其所需最小生成元数。
We show that the (2,4,5) triangle Coxeter group is not systolic.
Generalised Baumslag-Solitar groups and Hierarchically Hyperbolic Groups
本文研究任意维数广义Baumslag-Solitar群在双曲空间上的等距作用，证明实质上成为分层双曲群及性质（QT）在拟等距下均非不变量。
The generalised word problem for subgroups of hyperbolic groups
证明了双曲生成的虚自由群的广义拟凸子群的广义字问题是上下文无关的，证明了双曲群必是虚自由的，如果它有一个无穷指数的无挠拟凸子群，且其广义字问题是上下文无关的，并且证明了对于任意双曲群，无挠拟凸子群的广义字问题可由实时图灵机识别.
In this article, we completely characterize the asymptotic behavior of conjugacy separability for the lamplighter groups. More generally, we give exponential upper and lower bounds for all wreath products of finitely generated abelian groups where the acting group is infinite and base group is finite.
Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multi-edge spi…
We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…
The Tits alternative for non-spherical triangles of groups
三角形的群体已介绍了格斯滕和Stallings。粗略地说，它们是两个群的合并的自由积的概括，并且是在科森图的框架内发生的。首先证明了Corson图的一个交定理。然后，我们将重点放在三角形的群体。Howie和Kopteva证明了群的双曲三角形的上极限包含一个非阿贝尔自由子群。我们给出了两个自然条件，每一个条件都保证了群的非球面三角形的上极限包含一个非交换自由子群或者是虚拟可解的。
We show that a triangle Artin group $\text{Art}_{MNP}$ where $M\leq N\leq P$ splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$, or $N>3$. We also prove that all even three generator Artin groups are residually finite.
Triangle-free cyclic conjugacy class graph of a finite group
本文通过将元素替换为共轭类来推广增强幂图，并确定了该图无三角形的条件。
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, i.e. groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitiv…
Rank and deficiency gradients of generalised Thompson groups of type F
对广义Thompson群$$F_{n, \infty} = \langle x_0, x_1, \ldots, x_m, \ldots \mid x_i^{x_j} = x_{i+ n-1} \hbox{ for } i > j \geq 0 \rangle$$中的有限指数子群的任意序列$(G_s)$，证明了当$[G : G_s]$趋于无穷大时，$\sup_{s \geq 1} d(G_s) < \infty$和$F_{n, \infty}$关于$(G_s)$的亏梯度为0.考虑了$n = 2$的更高维类似物。
We prove a generalization of a conjecture of C. Marion on generation properties of finite groups of Lie type, by considering geometric properties of an appropriate representation variety and associated tangent spaces.
Triangles of Baumslag-Solitar Groups
我们的主要结果是Baumslag-Solitar群的许多三角形坍缩为有限群，推广了Hirsch的一个著名例子。Baumslag-Solitar群的三角形表示一个有三个生成元的群，循环有序，每个生成元将前一个生成元的某个幂共轭到另一个幂。有六个参数，成对发生，我们表明，三角形未能发展时，其中一个参数划分其合作伙伴，除了少数特殊情况。此外，在相当一般的条件下，这个群是有限的，并且是<4类的可解群。即使在我们不能确定可展性的情况下，我们也得到了许多关于有限导数的信息。
Residual Finiteness of $A_{2,3,2n}$ Triangle Artin Groups
通过将 $A_{2,3,2n}$ 型三角形 Artin 群分解为群图，并证明每个群图相对于其顶点群具有有限高度，证明了所有 $n\geq4$ 时这些群是剩余有限的。
Twisted Conjugacy Growth of the Generalised Heisenberg Groups
该研究确定了广义Heisenberg群自同构的扭曲共轭增长函数形式，通过上下中心序列的维数与秩区分两类增长形式，并证明扭曲共轭增长不超过共轭增长。
Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$.
Low rank groups of Lie type acting point and line-primitively on finite generalised quadrangles
本文证明有限厚广义四边形自同构群若在点和线上本原作用，则其基座不能为某些低秩李型群。
Let $V_r(Σ)$ be the generalised Thompson group defined as the automorphism group of a valid, bounded, and complete Cantor algebra. We show that that for every $n>0$ there is a $k>n,$ such that there exists a $k$-dimensional $n$-connected simplicial complex $K$ such that $V_r(Σ)$ acts on $K$ with finite stabilisers. We also determine the number of conjugacy classes of finite cyclic subgroups of a given order $m$ in Brin-Thompson groups. We apply our computations to the rationalised Farrell-Jones assembly map in algebraic $K$-theory.
Fixed subgroups of generalised Baumslag-Solitar groups
本文研究广义Baumslag-Solitar群的自同构固定子群，刻画了在边稳定子严格包含于顶点稳定子条件下存在非有限生成固定子群的群，并证明有限阶自同构及树图情形下固定子群均有限生成。
Motivated by the Problem $164$ proposed by Y. Berkovich and E. Zhmud' in their book "Characters of finite groups, Part $1$", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character coedgree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classif…
On Shephard Groups with Large Triangles
Shephard群是Artin群和Coxeter群的常见推广。例如，它们出现在流形的代数研究中。考虑一类非Artin或Coxeter群的Shephard群的无限族。使用技术形成小的取消理论，我们表明，在这个家庭中的群体是双自动的。
Quotient order density of ordinary triangle groups
本文利用Bertram定理启发的方法和Turan-Kubilius不等式，证明了任意普通三角群的有限商阶在正整数中的自然密度为零，回答了Larson等人提出的问题。
Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "m…
Deformation theory and finite simple quotients of triangle groups II
本文是我们的第一篇文章[10]的继续，在这篇文章中，我们展示了如何利用表示簇的变形理论来研究三角形群的有限单直子。而在第一部分，我们主要使用变形的主同态从${\rm SO}(3,\R)$，在这一部分中，我们使用${\rm PGL}_2(\R)$以及变形的表示，这是非常不同的主同态。
A triangle group is denoted by $Δ(p,q,r)$ and has finite presentation $$ Δ(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $Δ(p,q,r)$ that was used in Everitt's proof of Higman's 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations, and in particular situations, where the representations being composed are all equivalent copies of an alternating group in the same de…
Pascal's Triangles in Abelian and Hyperbolic Groups
帕斯卡三角形将给出从恒等式到${\bf Z}^2$的每个点的测地线的数量，如果你把它写在每个象限中。给定一个群$G$和生成集$\cal G$，我们取{ \it Pascal函数} $p_{\cal G}: G \to {\bf Z}_{\ge 0}$作为将从$1$到$g$的测地线的数目分配给每个$g\in G$的函数。我们给出了一个一般的方法来计算这在双曲群和讨论一般情况下，在阿贝尔群。
On generation problems in generalised Wilson type groups
本文研究了由J.Wilson在2010年引入的迭代圈积得到的一族遗传公正无限投射有限群.我们发现这个家庭在一些情况下，使用组合方法明确的发电机。然后，我们讨论确定较低的职级为这个家庭。最后，我们研究是否在这个家庭的群体承认有限profinite介绍。
Golan and Sapir \cite{MR3978542} proved that the Thompson's groups $F$, $T$ and $V$ have linear divergence. In the current paper, we focus on the divergence properties of several generalisation of the Thompson's groups, we first consider the Brown-Thompson's groups $F_n$, $T_n$ and $V_n$ and found out that these groups also have linear divergence function. We then consider the braided Thompson's groups $BF$ and $\widehat{BF}$ and $\widehat{BV}$ together with the result in \cite{Kodama:2020to} we conclude that theses groups have linear divergence.
Groups with triangle-free graphs on p-regular classes
该研究针对素数p，对满足“任三个非中心p-正则元共轭类中必有两类长度互素”的有限p-可分群的p-结构展开分类，推进了群论中相关结构问题的研究。
Soluble quotients of triangle groups
该研究通过证明非完满双曲普通三角群存在导出长度不超过3的光滑有限可解商群且更大导出长度的此类商群有无穷多，解释了小亏格正则地图自同构群中可解群普遍存在的原因。
The Tits alternative for generalized triangle groups of type (3,4,2)
广义三角形群是可以以$G = < x,y | x^p=y^q=w(x,y)^r=1>$形式表示的群，其中$p,q,r\geq 2$和$w(x,y)$是自由积$\Z_p*\Z_q=< x,y | x^p=y^q=1>$中长度至少为2的循环约简字。Rosenberger证明了每个广义三角形群$G$都满足Tits择一性。已知该猜想成立，除非三元组$(p,q,r)$可能是$(2,3,2), (2,4,2),(2,5,2),(3,3,2),(3,4,2)$或$(3,5,2)$之一时。在本文中，我们表明，山雀替代持有的情况下$(p,q,r)=(3,4,2)$。
On the biautomaticity of CAT(0) triangle-square groups
本研究针对CAT(0)三角-平方群的双自动性问题，构造两个反例说明Gersten-Short测地线无法用于证明相关群的双自动性，同时证明了其中一个复形基本群的双自动性。
Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type
设$G$是厚有限广义六边形或广义八边形$Γ$的直射变换群.如果$G$在$Γ$的点上连续作用，则Bamberg等人的一个最近结果表明，$G$必定是一个几乎单群.此外，我们表明，$G$的最小正规子群$S$不能是一个铃木群或一个$^2\text{G}_2$型的Ree群，并且如果$S$是一个$^2\text{F}_4$型的Ree群，则$Γ$是（直到点线对偶）经典的Ree-Tits广义八角形。
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.
We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow $2$-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow $2$-subgroups realised by Scott modules.
A generalised quadrangle is a point-line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits, and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as au…
A Beauville group acts freely on the product of two compact Riemann surfaces and faithfully on each one of them. In this paper, we consider higher products and present {\it{generalised Beauville groups}}: for $d \geq 2$, $d$ is the minimal value for which the same action can be defined on the product of $d$ compact Riemann surfaces.
Automorphisms of Cayley graphs on generalised dicyclic groups
一个图称为GRR，如果它的自同构群规则地作用在它的顶点集上。这样的图必然是凯莱图。Godsil表明，只有两个无限家庭的有限团体不承认GRR：阿贝尔群和广义双循环群。事实上，这样一个群上的任何凯莱图都允许只依赖于群的特定的附加图自同构。最近，多布森和最后两位作者表明，几乎所有的凯莱图阿贝尔群承认没有自同构以外的这些明显的必要。在本文中，我们证明了类似的结果凯莱图的其余家庭的例外群：广义双循环群。
We extend the result of \cite{Dutra} to generating pairs of triangle groups, that is, we show that any generating pair of a triangle group is represented by a special almost orbifold covering.
We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $\mathcal{S}$ is a finite thick generalised hexagon or octagon with $G \leqslant{\rm Aut}(\mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${\rm PSL}_n(q)$ where $n \geqslant 2$, then the stabiliser of a point acts irreducibly on the…
Arithmeticity of complex hyperbolic triangle groups
复双曲三角形群首先由Mostow在PU（2，1）中建立了第一个非算术格。它们是作用在双曲平面上的经典三角形群的自然推广。Takeuchi的一个著名定理是PSL_2（R）中同时是算术格的Fuchsian三角群只有1000个。我们考虑复双曲三角群的相似有限性定理。特别地，我们证明了在PU（2，1）中有许多具有有理角不变量的复双曲三角形群决定了一个算术格。我们还证明有限时，三角形是一个权利或等边三角形，后者的情况下是一个吸引了最大的关注，因为莫斯托的原始工作。
Point-primitive generalised hexagons and octagons
在2008年，Schneider和Van Maldeghem证明了，如果一个群在广义六边形或广义八边形上的作用是旗传递的、点传递的和线传递的，那么它是几乎单群的李型。我们表明，点介电性是足够的相同的结论，无论线或标志的行动。这一结果缩小了搜索广义六边形或八角形与点或线本原直射变换群超越经典的例子，即两个广义六边形和一个广义八角形承认李型群$\mathsf{G}_2(q)$，$\,^3\mathsf{D}_4(q)$，和$\,^2\mathsf{F}_4(q)$，分别。
Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type - the details
我们证明，广义非交叉分区以及生成的复杂的反射群的例外类型服从两个不同的循环筛分现象，如阿姆斯特朗，分别由贝西斯和赖纳。这份手稿伴随着论文“Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type”[arXiv：1001.0028]，它提供了计算细节。
P-adic Schwarzian triangle groups of Mumford type
本文讨论了经典Schwarzian三角形群的一个p-adic类比，这个类比与Mumford的p-adic解析曲线的一致化有关。定义p-adic Schwarzian三角形群为射影直线上具有恰好3个分支点的解析覆盖的伽罗瓦群.我们说一个p-adic三角群是Mumford型的，如果对应的泛覆盖是由射影直线上的某个局部紧解析子空间给出的，这与Mumford的单值化有关。主要定理提供了一个完整的分类p-adic三角群的芒福德型;我们的名单，这些团体包含一些算术p-adic三角群讨论了伊夫安德烈。值得注意的是，我们推导出Mumford型的p-adic三角群仅当p= 2，3，5时才存在。
The Action of Generalised Symmetric Groups on Symmetric and Exterior Powers of Their Natural Representations
我们建立了广义对称群的自然特征的对称幂所张成的空间的维数的下界。我们采用Savitt和Stanley的方法，从他们的论文“A note on the symmetric powers of the standard representation of $S_n$”（The Electronic Journal of Combinatorics 7（2000），R6）中获得复数和素数特征的界。
Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type
我们提出了证明的循环筛分的广义非交叉分区相关的良好生成的复杂的反射群由于阿姆斯特朗，分别贝西斯和雷纳，为26个特殊的良好生成的复杂的反射群。计算细节在手稿“与特殊类型的复杂反射群相关的广义非交叉分区的循环筛选-细节”[arXiv：1001.0030]中提供。
The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of…
In this paper we provide new examples of hyperbolic but nonsystolic groups by showing that the triangle groups $(2,4,5)$ and $(2,5,5)$ are not systolic. Along the way we prove some results about subsets of systolic complexes stable under involutions.
On generalised Paley graphs and their automorphism groups
广义佩利图，顾名思义，是佩利图的推广，佩利图首先在1933年定义（见\cite{Paley}）。它们作为对称分圆结合方案的关系图而出现。然而，它们的自同构群可能比相应方案的群大得多。我们确定的参数，图是连接的，或等价地，该计划是原始的。此外，我们证明，广义Paley图有时同构的汉明图，因此有大的自同构群，我们精确地确定参数发生。证明了在连通的非Hamming情形下，广义Paley图的自同构群是仿射型本原群，并给出了该群等于伴随分圆结合方案的一维仿射群的充分条件.这些结果已被应用在\cite{LLP}区分分圆计划和类似的扭曲版本的这些计划，在完全图的齐次因式分解的上下文中。
Generalised Cartan invariants of symmetric groups
Külshammer、Olsson和Robinson发展了对称群的模表示理论的l-模拟，其中l不一定是素数。他们给出了Cartan矩阵的不变因子的一个代数组合描述。我们证实了他们的猜想证明一个更精确的blockwise猜想由于Bessenrodt和希尔。
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