Alexander Kolpakov, Alexey Talambutsa
We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal $1$. Also, we compute the average number of geodesics representing an element of given word length in such groups.
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Geodesic growth in right-angled and even Coxeter groups
本文的目的是检测一个正则图的哪些组合性质可以完全确定直角Coxeter或Artin群的测地增长图定义，并提供直角和甚至Coxeter群具有相同的测地增长系列的第一个例子。
We determine the asymptotic number of index $n$ subgroups in virtually cyclic right-angled Coxeter groups and their free products as $n\to\infty$.
Recognizing Right-Angled Coxeter Groups Using Involutions
我们考虑的问题，确定是否一个给定的组（特别是一个生成的对合）是一个直角Coxeter群。我们描述了一个群不变量，对合图，我们的特征的直角Coxeter群的对合图。我们用这个特征来描述一个过程，为一个给定的组或证明，一个不可能存在的候选人直角Coxeter介绍。我们提供一些初步的应用。此外，我们还给出了直角Coxeter群定义图的刚性的一个初等证明。我们还恢复了一个结果，即如果定义图不包含SIL，则Aut^0（W）是直角Coxeter群。
We show that the Aut-invariant word norm on right angled Artin and right angled Coxeter groups is unbounded (except in few special cases). To prove unboundedness we exhibit certain characteristic subgroups. This allows us to find unbounded quasimorphisms which are Lipschitz with respect to the Aut-invariant word norm.
We establish quasi-isometric rigidity for a class of right-angled Coxeter groups. Let $Γ_1,Γ_2$ be joins of finite generalized thick $m$-gons with $m\geq 3$. We show that the corresponding right-angled Coxeter groups are quasi-isometric if and only if $Γ_1,Γ_2$ are isomorphic. We also give a construction of commensurable right-angled Coxeter groups.
Rigidity of Right-Angled Coxeter Groups
研究直角Coxeter群的刚性性质，证明不同生成集下的Coxeter系统等价性。
We study Morse subgroups and Morse boundaries of random right-angled Coxeter groups in the Erdős--Rényi model. We show that at densities below $\left(\sqrt{\frac{1}{2}}-ε\right)\sqrt{\frac{\log{n}}{n}}$ random right-angled Coxeter groups almost surely have Morse hyperbolic surface subgroups. This implies their Morse boundaries contain embedded circles and they cannot be quasi-isometric to a right-angled Artin group. Further, at densities above $\left(\sqrt{\frac{1}{2}}+ε\right)\sqrt{\frac{\log{n}}{n}}$ we show that, almost surely, the hyperbolic Morse special subgroups of a random right-angled…
In this article, we study the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.
Actions of Right-Angled Coxeter Groups on the Croke Kleiner Spaces
这是一个悬而未决的问题是否直角Coxeter群有唯一的组等变视觉边界。克罗克和克莱纳提出了一个直角的Artin组，具有多个视觉边界。本文给出了一个具有非唯一等变视觉边界的直角Coxeter群。主要定理是，如果直角Coxeter群几何地作用在Croke-Kleiner空间上，则这些空间中的局部角都必须是直角。我们提出了一个特殊的直角Coxeter群与非唯一的等变视觉边界。然而，我们猜想，直角Coxeter群，可以几何作用在一个给定的CAT（0）空间是远远不是唯一的。
Relative hyperbolicity and right-angled Coxeter groups
我们发现，直角Coxeter群是相对双曲的意义上定义的Farb，相对于自然收集的秩2抛物子群。
Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups
本文研究了有限图、单纯旗复形和直角Coxeter群之间的关系，给出了一类可重构的有限图。证明了如果$Γ$是一个有限图，且该图是某个单旗复形$L$的1-骨架，而$L$是一个维数为$n \ge 1$的同调流形，则图$Γ$是可重构的.
We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle-free defining graphs such that the associated right-angled Coxeter group has boundary a Menger curve.
$\mathrm{CAT}(0)$ Extensions of Right-angled Coxeter Groups
证明了直角Coxeter群$W_Γ$的任意有穷级生成自同构的分裂扩张忠实地几何地作用在$\mathrm{CAT}(0)$度量空间上.
RAAGedy right-angled Coxeter groups II: in the quasiisometry class of the tree RAAGs
本文对拟等距于树 RAAG 的二维右角 Coxeter 群分类，证明满足条件的此类群含可见有限指数右角 Artin 子群，解决了特定拟等距类群的结构问题。
Reflection subgroups of odd-angled Coxeter groups
本文给出了一个由反射生成的奇角Coxeter群具有一个由反射生成的真有限指数子群的判据。答案是根据最小的素除子的指数的考克斯特关系。
Divergence in right-angled Coxeter groups
设W是二维直角Coxeter群。我们用线性和二次发散度证明了这样的W，并构造了具有任意次发散多项式的直角Coxeter群。我们的证明使用的结构墙在戴维斯复杂。
We show that the Morse boundary of a right-angled Coxeter group may contain embedded circles that do not arise as the boundary of a Morse Fuchsian subgroup visible in the defining graph.
This is a survey of some aspects of the large-scale geometry of right-angled Coxeter groups. The emphasis is on recent results on their negative curvature properties, boundaries, and their quasi-isometry and commensurability classification.
The strong Atiyah conjecture for right-angled Artin and Coxeter groups
证明了直角Artin群和直角Coxeter群的强Atiyah猜想。更一般地说，我们证明了它的某些有限扩张或基本服从扩展的群体，这样的群体。
A classification of right-angled Coxeter groups with no 3-flats and locally connected boundary
若$(W,S)$是直角Coxeter系统，且$W$没有$\mathbb Z^3$子群，则证明了$(W,S)$的表示图中没有初等分离性意味着所有被$W$几何作用的CAT（0）空间都有局部连通的CAT（0）边界.如果一般直角Coxeter系统的表示图满足分离性，则$W$几何作用的所有CAT（0）空间都有非局部连通边界.特别地，这给出了没有3-平坦且具有局部连通边界的直角Coxeter群的完全分类。
Strongly Primitive Salem Growth Polynomials for Right-Angled Coxeter Groups
通过定义图的团多项式研究直角Coxeter群的标准球面增长率，证明每个偶数度至少4可作为强本原Salem增长率的度数，并给出独立多项式对指定Salem多项式的阻碍。
We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.
In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $π/2$ angles between reflecting axes, while the quasi-isometric right-angled Artin group can act with angles that are any real number in the range $(0, π/2]$. The contrast between the two examples shows that in this case a right-angled Coxeter group is geometrically more "rigid" than its quasi isometric cou…
On asymptotic dimension of amalgamated products and right-angled Coxeter groups
证明了不等式$$ \as A\ast_CB\le\max\{\as A,\as B,\as C+1\} $$，并应用这个不等式证明了任意直角Coxeter群的渐近维数不超过其Davis复形的维数.
A Perron theorem for matrices with negative entries and applications to Coxeter groups
Handelman（J. Operator Theory，1981）证明了如果矩阵$A$的谱半径是特征多项式的单根并且严格大于任何其他根的模，则$A$共轭于矩阵$Z$，其中某些幂是正的。本文给出了一个显式共轭矩阵$Z$，并证明了$A$的谱半径是$A$的单优势特征值当且仅当$Z$最终为正.对于每一行和都等于$1$的$n\times n$实矩阵，这个准则可以归结为检查某个幂的每一项严格大于同一列的项的平均值减去$\frac{1}{n}$。我们应用该准则的元素不可约无限非仿射Coxeter群的谱半径，这仍然是未知的优势提供证据。
Minimality of the boundary of a right-angled Coxeter system
本文证明了直角Coxeter系统$(W,S)$的边界$\partialΣ(W,S)$是极小的当且仅当$W_{\tilde{S}}$是不可约的，其中$W_{\tilde{S}}$是$W$中有限指数的极小抛物子群.我们还提供了几个应用程序和注释。特别地，我们得到了对于直角Coxeter系统$(W,S)$，集合$\{w^{\infty} | w\in W, o(w)=\infty\}$在边界$\partialΣ(W,S)$是稠密的.
We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group's defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group's order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n>1, we give examples of right-angled Coxeter groups which are thick o…
Coxeter systems with $2$-dimensional Davis complexes, growth rates and Perron numbers
本文研究低维Coxeter系统的增长率，证明其神经欧拉示性数的符号对应增长率为Salem、Pisot数，并构造了负欧拉示性数下增长率为Perron数的无穷多非双曲Coxeter系统，推广了已有结果。
We define a large class of abstract Coxeter groups, that we call $\infty$--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We also show that for this class the geodesic growth rate strictly dominates the word growth rate.
We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb H^4$ with fundamental domain the $120$-cell or the $24$-cell.
We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.
In this paper, we prove that all finitely generated malnormal subgroups of one-ended right-angled Coxeter groups are strongly quasiconvex and they are in particular quasiconvex when the ambient groups are hyperbolic. The key idea is to prove all infinite proper malnormal subgroups of one-ended right-angled Coxeter groups are join-free and then prove the strong quasiconvexity and the virtual freeness of these subgroups. We also study the subgroup divergence of join-free subgroups in right-angled Coxeter groups and compare them with the analogous subgroups in right-angled Artin groups. We charac…
The lower central series of the rgiht-angled Coxeter group $RC_\mathcal K$ and the corresponding graded Lie algebra $L(RC_\mathcal K)$ associated with the lower central series of a right-angled Coxeter group are studied. Relations are obtained in the graded components of the Lie algebra $L(RC_\mathcal K)$. A basis of the fourth graded component $L(RC_\mathcal K)$ for groups with at most $4$ generators was described.
We study the lower central series of a right-angled Coxeter group $RC_K$ and the associated Lie algebra $L(RC_K)$. The latter is related to the graph Lie algebra $L_K$. We give an explicit combinatorial description of the first three consecutive factors of the lower central series of the group $RC_K$.
Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group $\mathrm{O}(p, q)$ for some $p, q \geq 1$ satisfying $p+q=n$.
We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. As another application, we provide an inductive graph theoretic criteria on a right-angled Coxeter group's defining graph which allows us to recognize arbitrary i…
On consecutive factors of the lower central series of right-angled Coxeter groups
该研究针对右角Coxeter群的下中心列及相伴分次Lie代数展开，核心是描述任意$\u212c$对应的第四分次分量的基，为相关代数结构分析提供基础。
Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups
证明了有限阿贝尔群的图积初等等价当且仅当它们是$\exists\forall$ -等价的当且仅当它们是同构的.特别地，两个直角Coxeter群初等等价当且仅当它们同构。
Let $Γ$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $Γ$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_Γ$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_Γ$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_Γ$. Consequently, the divergence of $G_Γ$ is linear, or quadratic, or exponential. We also generalize right-angled …
Equivariant formality and the cohomology of subgroups of right-angled Coxeter groups
本文研究了右角Coxeter群的coabelian子群的分类空间模型，通过实moment-angle复形的同调轨道空间构造，并利用Borel等效上同调理论分析其上同调结构，提出图论判据判断coabelian子群的上同调自由性。
In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq 17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $T^1_n$ and $T^2_n$ have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree does is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $\{S^1_n\}_{n \geq 11}$ and $\{S^2_n\}_{n \geq 11}$, on $n$ vertices, such that $S^1_n$ and…
Higman-Thompson groups and profinite properties of right-angled Coxeter groups
本文证明了右角Coxeter群在Coxeter群中具有profinite刚性，并展示了某些RACG具有无限profinite genus，同时推广了Higman关于Thompson群$V$的生成结果。
Local Connectivity of Right-angled Coxeter group boundaries
本文研究直角 Coxeter 群所作用 CAT(0) 空间边界的局部连通性，给出由定义图刻画的充分条件，并对已发表版本进行证明精简与插图补充。
Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups
本文研究二维直角 Coxeter 群中直角 Artin 有限指数子群的存在性，将其与图的卫星可拆解性联系起来，并给出构造合适子图或判定不存在的算法。
We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only if $\mathscr{K}$ is either a $p$-cycle with $p\geqslant4$ or has the form ($p$-cycle)$*Δ^q$ with $p\geqslant4$ and $g\geqslant0$, where $Δ^q$ is a $q$-simplex.
Right-angled Artin subgroups of right-angled Coxeter and Artin groups
本文研究了直角Coxeter群和直角Artin群的子群何时为RAAGs，刻画了2维RACGs的有限指数视觉RAAG子群，并应用此结果证明了2维单端RACG拟等距于RAAG的充要条件，同时给出了非平面定义图的新例子及Dyer结果的新证明。
A threshold for relative hyperbolicity in random right-angled Coxeter groups
本文确定了以Erd{\H o}s--Rényi随机图为表现图的随机右角Coxeter群相对双曲性的阈值为$p=1/\sqrt{n}$，并揭示了其发散性区间及对应随机游走的中心极限定理性质。
Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös-Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about the homology of the nerve of a random Coxeter group and results about when random Coxeter groups are $δ$-hyperbolic and when they have the FC-type property.
Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups
本文证明 Nica 关于边界存在无不动点对合的猜想在直角 Coxeter 群情形下的变体，并为原词双曲群问题提供方法启发。
In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that, if two right-angled Coxeter groups $C(Γ_1)$ and $C(Γ_2)$ are quasi-isometric, then for any minsquare subgraph $Λ_1 \leq Γ_1$ there exists a minsquare subgraph $Λ_2 \leq Γ_2$ such that the right-angled Coxeter groups $C(Λ_1)$ and $C(Λ_2)$ are quasi-isometric as well. Various examples of non-quasi-isometric groups are deduced. Our arguments are based on a stu…
We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter at most 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group w…
We completely classify the possible divergence functions for right-angled Coxeter groups (RACGs). In particular, we show that the divergence of any such group is either polynomial, exponential or infinite. We prove that a RACG is strongly thick of order k if and only if its divergence function is a polynomial of degree k+1. Moreover, we show that the exact divergence function of a RACG can easily be computed from its defining graph by an invariant we call the hypergraph index.
This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition.
To a Coxeter group $W$ one associates a quandle $X_W$ from which one constructs a group $\mathrm{Ad}(X_W)$. This group turns out to be an intermediate object between $W$ and the associated Artin group. By using a result of Akita, we prove that $\mathrm{Ad}(X_W)$ is given by a pullback involving $W$, and by using this pullback, we show that the classifying space of $\mathrm{Ad}(X_W)$ is given by a space called a polyhedral product whenever $W$ is right-angled. Two applications of this description of the classifying space are given.
In this article, given two finite simplicial graphs $Γ_1$ and $Γ_2$, we state and prove a complete description of the possible morphisms $C(Γ_1) \to C(Γ_2)$ between the right-angled Coxeter groups $C(Γ_1)$ and $C(Γ_2)$. As an application, assuming that $Γ_2$ is triangle-free, we show that, if $C(Γ_1)$ is isomorphic to a subgroup of $C(Γ_2)$, then the ball of radius $8|Γ_1||Γ_2|$ in $C(Γ_2)$ contains the basis of a subgroup isomorphic to $C(Γ_1)$. This provides an algorithm determining whether or not, among two given two-dimensional right-angled Coxeter groups, one is isomorphic to a subgroup o…
We generalise the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph $Γ$ of a right-angled Coxeter group $W_Γ$ so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group $F_2$ as a quotient. When $Out(W_Γ)$ is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite, or equivalently $Γ$ …
On the Weak Right Order of a Right-Angled Coxeter System
本文证明在任意Coxeter系统中，所有非单位完全交换元素具有祖先性质，特别地，直角Coxeter系统的所有非单位元素满足该性质，并给出了直角Coxeter系统的公理化刻画。
We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgro…
Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups
与参数$q$相关联的Coxter群$W$的Hecke代数$\mathbb{C}_q[W]$可以被完成为von Neumann代数$\mathcal{N}_q(W)$。我们研究了$W$是直角的情况下的代数。我们确定$q$的范围，其中$\mathcal{N}_q(W)$是因子，即具有平凡中心。此外，在非平凡中心的情况下，我们证明了一个结果，允许分解$\mathcal{N}_q(W)$到一个有限的直和的因素。
Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ γ_1, \ldots , γ_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group such that $f (γ_i) \notin f (H)$. A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense. Similarly, finitely gene…
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