Matching theorems for systems of a finitely generated Coxeter group

Reflection subgroups of odd-angled Coxeter groups

We give a criterion for a finitely generated odd-angled Coxeter group to have a proper finite index subgroup generated by reflections. The answer is given in terms of the least prime divisors of the exponents of the Coxeter relations.

Reflection centralizers in Coxeter groups

We refine Brink's theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for centralizer, which is finitely generated when W is. And we give a method for computing the Coxeter diagram for its reflection subgroup. In many cases, our method allows one to compute centralizers in one's head.

Garside families in Artin-Tits monoids and low elements in Coxeter groups

We show that every finitely generated Artin-Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.

Coxeter Groups are not higher rank Arithmetic Groups

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits form of the Coxeter group is non-positive and non-degenerate, the Coxeter group does not contain any finite index subgroup isomorphic to an irreducible lattice in a semisimple group of R-rank greater or equal to 2.

On the maximally clustered elements of Coxeter groups

We continue the study of the maximally clustered elements for simply laced Coxeter groups which were recently introduced by Losonczy. Such elements include as a special case the freely braided elements of Losonczy and the author, which in turn constitute a superset of the $iji$-avoiding elements of Fan. Our main result is to classify the MC-finite Coxeter groups, namely those Coxeter groups having finitely many maximally clustered elements. Remarkably, any simply laced Coxeter group having finitely many $iji$-avoiding elements also turns out to be MC-finite.

Freely braided elements in Coxeter groups, II

We continue the study of freely braided elements of simply laced Coxeter groups, which we introduced in a previous work (math.CO/0301104). A known upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is shown to be achieved only when the element is freely braided; this establishes the converse direction of a previous result. It is also shown that a simply laced Coxeter group has finitely many freely braided elements if and only if it has finitely many fully commutative elements.

Virtually splitting the map from Aut(G) to Out(G)

We give an elementary criterion on a group G for the map from Aut(G) to Out(G) to split virtually. This criterion applies to many residually finite CAT(0) groups and hyperbolic groups, and in particular to all finitely generated Coxeter groups. As a consequence the outer automorphism group of any finitely generated Coxeter group is residually finite and virtually torsion-free.

On the non-discreteness of automorphism groups of Cayley graphs of Coxeter groups

In this work we characterise Cayley graphs of Coxeter groups with respect to the standard generating set that admit uncountable vertex stabilisers. As a corollary, we fully identify finitely generated Coxeter groups for which the automorphism group of their Cayley graph with respect to the standard generating set is not discrete when equipped with the permutation topology. As an application, we also provide new explicit constructions of vertex-transitive graphs of infinite degree that have locally compact automorphism groups.

Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

Let $(W, S)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $(W, S)$ for $W$ to be relatively hyperbolic with respect to a collection of finitely generated subgroups. The peripheral subgroups are necessarily parabolic subgroups (in the sense of Coxeter group theory). As an application, we present a criterion for the maximal flats of the Davis complex of $(W,S)$ to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type $(W,S)$ are isolated.

CAT(0) cubical complexes for graph products of finitely generated abelian groups

We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it is related to the embedding into a right-angled Coxeter group described by Davis and Januszkiewicz. We compare the approaches and also adapt the argument that the action extends to finite index supergroup that is a graph product of finite groups.

Virtually Fibering Right-Angled Coxeter Groups

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.

Simplicity of the reduced C-*-algebras of certain Coxeter groups

Let (G,S) be a finitely generated Coxeter group, such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced C-*-algebra of G is simple with unique normalised trace. For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants C>0 and a natural number L such that for a function f in l^2(G) supported on elements of length n with respect to the generating set S: || f * h || <= C(n+1)^{3/2 L} || f || || h ||, forall h in l^2(G).

Twist-rigid Coxeter groups

We prove that two angle-compatible Coxeter generating sets of a given finitely generated Coxeter group are conjugate provided one of them does not admit any elementary twist. This confirms a basic case of a general conjecture which describes a potential solution to the isomorphism problem for Coxeter groups.

Strong accessibility of Coxeter groups over minimal splittings

Given a class of groups C, a group G is strongly accessible over C if there is a bound on the number of terms in a sequence L(1), L(2), ..., L(n) of graph of groups decompositions of G with edge groups in C such that L(1) is the trivial decomposition (with 1-vertex) and for i>1, L(i) is obtained from L(i-1) by non-trivially and compatibly splitting a vertex group of L(i-1) over a group in C, replacing this vertex group by the splitting and then reducing. If H and K are subgroups of a group G then H is smaller than K if H intersect K has finite index in H and infinite index in K. The minimal sp…

Decompositions de groupes par produit direct et groupes de Coxeter

We provide examples of groups which are indecomposable by direct product, and more generally which are uniquely decomposable in direct products of indecomposable groups. Examples include Coxeter groups, for which we give an alternative approach to recent results of L. Paris. For a finitely generated linear group Gamma, we establish an upper bound on the number of factors of which Gamma can be the direct product. If Gamma is moreover torsion-free, it follows that Gamma is uniquely decomposable in direct product of indecomposable groups.

Higman-Thompson群与右角Coxeter群的紧致性质

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。

Semisimple Zariski closure of Coxeter groups

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.

Neighbourhoods in root systems of infinite Coxeter groups

Let $W$ be a finitely generated infinite Coxeter group, with $Φ$ and $Π$ being the corresponding root system and set of simple roots respectively. It has been observed by Hohlweg et la that the projections of elements of $Φ$ onto suitably chosen hyperplanes, called \emph{normalized roots}, are contained in the convex hull of $Π$ (which is a compact set), and hence the set of all normalized roots may exhibit interesting asymptotical behaviours. In this paper we investigate the topology of the limit set of the normalized roots and demonstrate a natural system of neighbourhoods around each limit …

Embedding of hyperbolic Coxeter groups into products of binary trees and aperiodic tilings

We prove that a finitely generated, right-angled, hyperbolic Coxeter group can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of the group. As application we obtain certain strongly aperiodic tilings of the Davis complex of these groups.

Visual Decompositions of Coxeter Groups

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter group is a subgroup of the fundamental group of a given graph of groups, then the Coxeter group is also the fundamental group of a graph of special subgroups, where each vertex and edge group is a subgroup …

The low-dimensional homology of finite-rank Coxeter groups

We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.

Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.

Alternating quotients of right-angled Coxeter groups

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…

仿射不可约Coxeter群的强壳性质

The strong hull property for affine irreducible Coxeter groups of rank 3

本文证明了所有秩为3的仿射不可约Coxeter群满足强壳性质，通过Coxeter复形的几何性质将凸壳分析简化为有限可管理的配置。

On the difficulty of presenting finitely presentable groups

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of …

Malnormality and join-free subgroups in right-angled Coxeter groups

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…

On subgroups of Coxeter groups

A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are related to homological properties of the simplicial complex whose simplices are the sets of commuting generators. Using these techniques, we construct torsion-free groups which are Poincare duality groups over some rings but not over others, and a group whose integral cohomological dimension is finite but strictly greater than its cohomological dimension over…

关于 Coxeter 群中对合前缀的一个注记

A note on involution prefixes in Coxeter groups

本文研究了 Coxeter 群中对合前缀的祖先性质，证明了有限生成 Coxeter 群的所有 Coxeter 元均具有该性质并具有对合乘积的典范表达式，同时猜想此性质对有限 Coxeter 群的所有非单位元均成立。

Finite presentability of Kac-Moody groups over finite fields

Let $\mathcal{G}$ be a Kac-Moody group functor in the sense of Tits, with associated Coxeter system $(W,S)$. For any field $F$, the group $\mathcal{G}(F)$ is finitely generated iff $F$ is finite. We are interested in the question when $G = \mathcal{G}(\mathbb{F}_q)$ is finitely presented. If $(W,S)$ is 2-spherical, it is well known that this is "almost always" the case. It is conjectured that $G$ is never finitely presented if $(W,S)$ is not 2-spherical (which means that there exist $s,t \in S$ with $|st| = \infty$), which so far (to the best of our knowledge) has only been proved for the type…

Tits 锥的对偶与 Coxeter 群的支配性

The dual of the Tits cone and dominance of Coxeter groups

本文针对有限生成无限 Coxeter 群，引入 Tits 锥对偶中的反射过程，刻画其极限，并应用于支配偏序下根的无限序列研究。

Coxeter群的双曲模型与稳定性识别

Hyperbolic models and stability recognition for Coxeter groups

本文针对有限秩Coxeter系统，通过锥化Davis复形的宽抛物子复形构造双曲空间，证明该空间兼具稳定性识别与Morse识别性质，且Coxeter群在其上的作用给出其最大非阿基米德双曲结构。

某些直角考克斯特群的拟等距

Quasi-Isometries for certain Right-Angled Coxeter Groups

本文针对有限展示、单端、二维直角考克斯特群，基于定义图构造柱面JSJ树，证明其二端边稳定子与定义图不含特定细分K4等价，并利用结构不变量给出由定义图决定的拟等距不变量，在无刚性顶点情形下进一步细化为完全拟等距不变量。

Coxeter 和 Dynkin 图

Coxeter and Dynkin Diagrams

本文介绍了 Coxeter 和 Dynkin 图的分类作用，涵盖有限反射群、简单李群、以及与之相关的代数结构。

JSJ Decompositions of Coxeter Groups

The idea of "JSJ-decompositions" for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of 2-sided, incompressible tori that separate a given closed irreducible 3-manifold into pieces with strong topological structure. Sela introduced the idea of JSJ-decompositions for groups, an idea that has flourished in a variety of directions. The general idea is to consider a certain class X of groups and splittings of groups in X by groups in another class Y. E.g. Rips and Sela considered splittings …

删除序与Coxeter群

The Deletion Order and Coxeter Groups

本文研究有限生成Coxeter群的删除序，证明其为Bruhat序的加细，定义了对应范式并证明与从右到左字典序范式一致，还得到了其Artin性质及与有限Coxeter群最长元素的相互作用等结果。

Mapping the Davis complex into the imaginary cone

The study of the set of limit roots associated to an infinite Coxeter group was initiated by Hohlweg, Labbé and Ripoll and further developed by Dyer, Hohlweg, Péaux and Ripoll. The Davis complex associated to a finitely generated Coxeter group $W$ is a piecewise Euclidean CAT(0) space on which $W$ acts properly, cocompactly by isometries. The one skeleton of the Davis complex can be identified with the Cayley graph of $W$. In this paper we define a natural map from the Davis complex into the normalised imaginary cone of a based root system.

Coxeter群中的规范连接表示与连接不可约元素

Canonical Join Representations and Join-irreducible elements of Garside shadows in Coxeter groups

本文研究了Coxeter群中元素的规范连接表示和连接不可约元素的性质，并提出了确定元素锥型的高效计算方法。

Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

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…

Open subgroups of locally compact Kac-Moody groups

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.

Coxeter群与分裂晶体群的同质性

Homogeneity in Coxeter groups and split crystallographic groups

本文研究了Coxeter群和晶体群的同质性，证明了某些双曲群和Coxeter群的同质性，并构造了非同质的分裂晶体群。

The adjoint group of a Coxeter quandle

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of $Q_W$ is isomorphic to the commutator subgroup of $W$. Finally, the root system $Φ_W$ associated with a Coxeter group $W$…

The absolute order of a permutation representation of a Coxeter group

A permutation representation of a Coxeter group $W$ naturally defines an absolute order. This family of partial orders (which includes the absolute order on $W$) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on $W$ are investigated when $W$ is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of $W$ is defined.

Subregular $J$-rings of Coxeter systems via quiver path algebras

We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-$A_K$ of finite dimensional right modules of the algebra $A_K=K\otimes_\Z J_C$ over an algebraically closed field $K$ of characteristic zero. Our results include classifications of Coxeter systems for which mod-$A_K$ is semisimple, has finitely many …

Automata, reduced words, and Garside shadows in Coxeter groups

In this article, we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated Coxeter system (W,S). The definition of these automata is straightforward as it only requires the notion of weak order on (W,S) and the related notion of Garside shadows in (W,S), an analog of the notion of a Garside family. Then we discuss the relations between this class of automata and the canonical automaton built from Brink and Howlett's small roots. We end this article by providing partial positive answers to two conjectures: (1) …

扭转猜想与考克斯特群的同构问题

The Twist Conjecture and the Isomorphism Problem for Coxeter groups

该研究证明了考克斯特群的扭转猜想，结合前期工作解决了考克斯特群的同构问题，还得出考克斯特群的自同构群有限生成的推论，未使用人工智能开展研究。

Hilbert space compression and exactness for discrete groups

We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.

Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groups

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $\mathbb{Z}$ whose kernel has strong topological finiteness proper…

图形离散性、Coxeter 加倍与广义多边形

Graphical Discreteness, Coxeter Doublings and Generalized Polygons

本文证明在拟等距于右角Coxeter群的有限生成群类中，图形离散性等价于广义多边形的非柔性，并结合建筑刚性与度量分层给出分类及若干应用。

OGS canonical forms and exchange laws for the I and for the A-type Coxeter groups

We consider a generalization of the fundamental theorem of finitely generated abelian groups for some non-abelian groups, which is called OGS. First, we consider the dihedral group, which is a non-abelian extension of an abelian group by an involution. Then, we focus on a special case, where the abelian group is cyclic, which is the two-generated Coxeter group I{2}(m). We mention interesting connections between the reduced Coxeter presentation and a particular OGS canonical presentation, which we call the standard OGS canonical presentation. These connections motivate us to offer a generalizat…

Coxeter群与作用于建筑的群上的Hilbert变换

Hilbert transforms on Coxeter groups and groups acting on buildings

本文通过几何对象建立新模型，证明Coxeter群及作用于建筑的群上Hilbert变换满足Cotlar恒等式，从而得到$L_p$-有界性，并首次在具有性质(F$\mathbb{R}$)的群上实现该结果。

Deformations of convex real projective manifolds and orbifolds

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups.

Existence, covolumes and infinite generation of lattices for Davis complexes

Let $Σ$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $Σ$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice $Γ$, and an infinite family of uniform lattices with covolumes converging to that of $Γ$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $Γ$ is not finitely generated. Examples of $Σ$ to which ou…

Small roots, low elements, and the weak order in Coxeter groups

In this article we provide a new finite class of elements in any Coxeter system (W,S) called low elements. They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of (W,S). Our first main result is to show that they form a Garside shadow in (W,S), i.e., they contain S and are closed under join (for the right weak order) and by taking suffixes. These low elements are the key to prove that all finitely generated Artin-Tits groups have a finite Garside family. This result was announced in a note with P. Dehornoy (P. Dehornoy, M. Dyer, and C. Hoh…

Groups which are not properly 3-realizable

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has {\em pro-(finitely generated free) fundamental group at infinity} and {\em semi-stable ends}. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups.