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 defining relations of the mixed-sign Coxeter groups with the mentioned associated graph (either a line or a simple cycle) are squares or cubes of a product of conjugates of two generators of the mixed-sign Coxeter group and are strongly connected to the labels of the vertices of the associated graph.
We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4-cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non-simply laced cases, whe…
On growth types of quotients of Coxeter groups by parabolic subgroups
本文主要研究既不是有限也不是仿射的Coxeter群$W$。de la Harpe的一个著名结果断言，这样的群具有指数增长性。本文利用$W$的抛物子群和某类反射子群来研究$W$的代数。我们发现这些变量也有指数增长性。为了实现这一点，我们使用戴尔定理构造一个反射子群的$W$是同构的通用Coxeter群上的三个发电机。这些结果都是在$W$的Coxeter图是简单的约束下证明的，并对如何放宽这一约束作了一些评论。
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…
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 consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{ä}ger \cite{shmuck} and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in \cite{Dyer}. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free.
Beijing International Center for Mathematical Research Peking University, Beijing, China（北京大学北京国际数学研究中心）；School of Mathematics and Statistics University of Melbourne, VIC 3010, Australia（墨尔本大学数学与统计学院）；Department of Foundational Mathematics School of Mathematics and Physics Xi’an Jiaotong-Liverpool University, Suzhou, China（西交利物浦大学数学与物理学院基础数学系）
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…
We study the asymptotic behaviour of the statistic (des+ides) which assigns to an element w of a finite Coxeter group W the number of descents of w plus the number of descents of its inverse. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and generalises work of Chatterjee-Diaconis, Özdemir and Röttger.
We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. In that context we speak of the compactification and the boundary of the Coxeter group. As it turns out, the canonical action of the Coxeter group on its Cayley graph induces a natural action on the compactification and the boundary. From thi…
