Patrick Bahls
We compute Aut(W) for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in terms of the given presentation for the Coxeter group and admits an easy characterization of those groups W for which Out(W) is finite.
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Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups
本文研究Coxeter群及其自同构群中的正规子群，刻画了其中有限子群与不含非阿贝尔自由群的窄子群的结构。
In this paper, we consider the automorphism groups of the Cayley graph with respect to the Coxeter generators and the Davis complex of an arbitrary Coxeter group. We determine for which Coxeter groups these automorphism groups are discrete. In the case where they are discrete, we express them as semidirect products of two obvious families of automorphisms. This extends a result of Haglund and Paulin.
We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Γ$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_Γ)$ virtually surjects onto some infinite Coxeter group. In particular, the group ${\rm Aut}(W_Γ)$ is virtually indicable and therefore does not have Kazhdan's property (T).
We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.
In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph automorphisms in Coxeter groups, which are of independent interest, are also given.
Property $R_\infty$ for Artin and Coxeter groups whose outer automorphisms are length-preserving
该研究证明非球面Artin与Coxeter群的保长自同构具无限Reidemeister数，进而确立多类Artin、Coxeter群的$R_\infty$性质，还证明无限不可约非仿射强刚性Coxeter群的自同构群具acylindrically双曲性。
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.
An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for which all the exponents $m_{ij}$ are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs $\mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same multiset of finite exponents. In particular, each group in this family is isomorphic to a group that admits a Coxeter system whose associated labeled graph is a star shaped tree. We give the complete de…
By underlying the commutation relation in a right-angled Coxeter group W, we recover the fact that right-angled Coxeter groups are rigid and we describe the second subgroup of Aut(W) that appears in the decomposition of Aut(W) into a semi-direct product established by J. Tits in "Sur le groupe des automorphismes de certains groupes de Coxeter" (Journal of Algebra 113, (1988), 346-357). ----- En mettant l'accent sur la relation de commutation dans un groupe de Coxeter a angles droits W, on redemontre le fait que les groupes de Coxeter a angles droits sont rigides et on decrit le second sous-gro…
We study fixed point properties of the automorphism group of the universal Coxeter group Aut$(W_n)$. In particular, we prove that whenever Aut$(W_n)$ acts by isometries on complete $d$-dimensional CAT$(0)$ space with $d<\lfloor\frac{n}{2}\rfloor$, then it must fix a point. We also prove that Aut$(W_n)$ does not have Kazhdan's property (T). Further, strong restrictions are obtained on homomorphisms of Aut$(W_n)$ to groups that do not contain a copy of Sym(n).
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 $Γ$ …
A characterization of virtually cyclic outer automorphism groups of right-angled Coxeter groups
本文研究了右角Coxeter群的外自同构群的虚循环性质，通过分析生成特定部分共轭的有限指数子群，确定了SILs、STILs和FSILs的存在与否对代数性质的影响。
Automorphismes du groupe des automorphismes d'un groupe de Coxeter universel
本文利用自由积的Guirardel-Levitt外空间，确定了秩$n \geq 4$的泛Coxeter群的自同构群及外自同构群的外自同构群结构。
Symmetry and Rigidity of Star-Shaped Coxeter Systems
本文研究了具有星形有限Coxeter图的Coxeter群的自同构群，证明了每个自同构可分解为内自同构、图自同构以及三种额外类型：转置和两个部分共轭家族，并探讨了自然的短正合列，利用Moussong的超几何性准则证明这些群具有R_∞性质，并利用已知技术建立了这些群的刚性性质，解决了星形Coxeter系统的同构问题。
We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer Space as a model for Out(F_n), we prove that the natural combinatorial and topological model for their outer automorphism groups can \emph{not} be given an equivariant CAT(0) metric. This is particularly interesting as there are very few non-trivial examples of proving that a mod…
Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $φ$ be a weight function which is invariant under $G$. Let $φ_G$ denote the weight function on $W^G$ obtained by restriction from $φ$. The aim of this paper is to compare the ${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,φ)$ and to $(W^G,φ_G)$, provided that Lusztig's Conjectures hold.
This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank $n$, which is the free product $W_n$ of $n$ copies of $\mathbb{Z}/2\mathbb{Z}$. We prove that for $n\geq 5$ the natural map $\mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n))$ is an isomorphism and that every isomorphism between finite index subgroups of $\mathrm{Out}(W_n)$ is given by a conjugation by an element of $\mathrm{Out}(W_n)$.
We prove that numerous negatively curved simply connected locally compact polyhedral complexes, admitting a discrete cocompact group of automorphisms, have automorphism groups which are locally compact, uncountable, non linear and virtually simple. Examples include hyperbolic buildings, Cayley graphs of word hyperbolic Coxeter systems, and generalizations of cubical complexes, that we call even polyhedral complexes. We use tools introduced by Tits in the case of automorphism groups of trees, and Davis-Moussong's geometric realisation of Coxeter systems.
We show that any split extension of a right-angled Coxeter group $W_Γ$ by a generating automorphism of finite order acts faithfully and geometrically on a $\mathrm{CAT}(0)$ metric space.
We consider the Cayley graph ${\rm C}(W,S)$ of a Coxeter system $(W,S)$ and describe all maximal $2$-cliques in this graph, i.e. maximal subsets in the vertex set such that the distance between any two distinct elements is equal to $2$. As an application, we show that every automorphism of the half of Cayley graph is uniquely extendable to an automorphism of the Cayley graph if $|S|\ge 5$.
We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that a non-spherical and non-affine Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. We prove that a Coxeter group has a decomposition as a direct product of indecomposable groups, and that such a decomposition is unique up to a central automorphism and a permutation of the factors. We prove that a Coxeter group has a virtual decomposition as a direct product of strongly indecomposable groups, and that such a decomposition is unique up…
We show that word hyperbolicity of automorphism groups of graph products $G_Γ$ and of Coxeter groups $W_Γ$ depends strongly on the shape of the defining graph $Γ$. We also characterized those $Aut(G_Γ)$ and $Aut(W_Γ)$ in terms of $Γ$ that are virtually free.
In this paper we prove, without the finite rank assumption, that any irreducible Coxeter group of infinite order is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group, the centralizers of the normal subgroups that are generated by involutions. As a consequence, we show that the problem of deciding whether two general Coxeter groups are isomorphic, as abstract groups, is reduced to the case of irreducible Coxeter groups, without assuming the finiteness of the number of the irreducible components or their ranks…
In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism…
Realizing Non-Archimedean Polish Groups as Outer Automorphism Groups
通过小消去理论，构造可数离散群使得其外自同构群同构于任意给定的非阿基米德波兰群，并给出一个Borel映射。
Uniclass automorphisms of spherical buildings
本文引入Weyl子结构概念，通过固定结构刻画球面建筑的Uniclass自同构，并将其与Freudenthal--Tits幻方建立联系。
In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups.
We study the group of type-preserving automorphisms of a right-angled building, in particular when the building is locally finite. Our aim is to characterize the proper open subgroups as the finite index closed subgroups of the stabilizers of proper residues. One of the main tools is the new notion of firm elements in a right-angled Coxeter group, which are those elements for which the final letter in each reduced representation is the same. We also introduce the related notions of firmness for arbitrary elements of such a Coxeter group and $n$-flexibility of chambers in a right-angled buildin…
The main result of this paper describes the normalizer of a finite parabolic subgroup of a (possibly infinite) Coxeter group. We use this to compute the automorphism groups of some Lorentzian lattices and K3 surfaces.
We consider the class of those Coxeter groups for which removing from the Cayley graph any tubular neighbourhood of any wall leaves exactly two connected components. We call these Coxeter groups bipolar. They include both the virtually Poincare duality Coxeter groups and the infinite irreducible 2-spherical ones. We show in a geometric way that a bipolar Coxeter group admits a unique conjugacy class of Coxeter generating sets. Moreover, we provide a characterisation of bipolar Coxeter groups in terms of the associated Coxeter diagram.
H.S.M. Coxeter showed that a group $Γ$ is a finite reflection group of an Euclidean space if and only if $Γ$ is a finite Coxeter group. In this paper, we define {\it reflections} of geodesic spaces in general, and we prove that $Γ$ is a cocompact discrete reflection group of some geodesic space if and only if $Γ$ is a Coxeter group.
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.
We prove the even isomorphism theorem for Coxeter groups
On the language of palindromic reduced words in a Coxeter group
本文证明了Coxeter系统中回文约化词的语言正则性等价于群为有限Coxeter群与无限二面体群有限直积的条件。
We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial tec…
The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. In this paper we complete the calculation of the strong symmetric genus for each finite Coxeter group excluding the group E8.
We show that the (2,4,5) triangle Coxeter group is not systolic.
We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphism generated by partial conjugations is itself a graph product of cyclic groups providing its defining graph does not contain any separating intersection of links (SIL). In the case that all the cyclic groups are finite, this implies that the automorphism group is virtually CAT(0); it has a finite index subgroup which acts geometrically on a right-angled building.
We use probabilistic methods to prove that many Coxeter groups are incoherent. In particular, this holds for Coxeter groups of uniform exponent > 2 with sufficiently many generators.
We show that the standard generating set of a Coxeter group is of minimal cardinality provided that the non-diagonal entries of the Coxeter matrix are sufficiently large.
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 …
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。
In this note, we give a remark on the structure of centralizers of involutions in Coxeter groups.
We study coherence of graph products and Coxeter groups and obtain many results in this direction.
Groupes de Coxeter finis : centralisateurs d'involutions
本文逐案描述了有限Coxeter群中对合的中心化子。
In this work we study representations of certain Coxeter groups to obtain some properties of the corresponding reflection groups.
This paper considers a smaller version of the 26-node presentation of the Bimonster group.
We attach with every finite, involutive, nondegenerate set-theoretic solution of the Yang--Baxter equation a finite group that plays for the associated structure group the role that a finite Coxeter group plays for the associated Artin--Tits group.
We prove that Coxeter groups are biautomatic. From our construction of the biautomatic structure it follows that uniform lattices in isometry groups of buildings are biautomatic.
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.
It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces, then the local angles in those spaces all have to be right angles. We present a specific right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right an- …
If $A_Γ$ ($W_Γ$) is the Artin (Coxeter) group with defining graph $Γ$ we denote by $Sim(Γ)$ the number of vertices of the largest clique in $Γ$. We show that $asdimA_Γ\leq Sim(Γ)$, if $Sim(Γ)=2$. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that $asdimW_Γ\leq Sim(Γ)$ for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptot…
In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the group.
A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every cycle of length at least four has a cord. As an application, we construct counterexamples to two main conjectures concerning the isomorphism problem for Coxeter groups.
Profinite rigidity of affine Coxeter groups
本文证明了仿射Coxeter群在 profinite 意义下是刚性的。
For the noncrystallographic Coxeter groups of type $H$, we construct their Gröbner-Shirshov bases and the corresponding standard monomials.
In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing the standard geometric representation of the associated Coxeter group by concerning the labels of the vertices. The group which Hironaka get by that generalization is called mixed-sign Coxeter group. In this paper we classify the simply-laced mixed-sign Coxeter groups where the associated graph is either a line or a simple cycle. We show that all the defini…
Exponential growth rates of even Coxeter groups
证明在所有生成偶Coxeter群W的Coxeter系统中，唯一偶Coxeter系统实现最小指数增长率，通过分析Mihalik算法中伪换位操作不增加增长率。
We provide involutory symmetric generating sets of finitely generated Coxeter groups, fulfilling a suitable finiteness condition, which in particular is fulfilled in the finite, affine and compact hyperbolic cases.
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.
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