相关论文: Freely braided elements in Coxeter groups, II

Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer…

Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = A(W) be the associated hyperplane arrangement of W. Terao has shown that each such reflection arrangement A is free. There is the stronger…

Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set of twisted involutions relative to $\ast$ in $W$. In this paper we consider the…

We show the uniqueness and existence of the Euler form for a simply-laced generalized root system. This enables us to show that the Coxeter element for a simply-laced generalized root system is admissible in the sense of R.~W.~Carter. As an…

We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate…

Wirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots. They are encoded by (word) labeled oriented trees and, for that reason, are also called LOT presentations. These presentations…

This is the second part of the paper. Results of the first part about crossed modules are applied here to study of quantum groups in braided categories. Correct cross product in the class of quantum braided groups is built. Criterion when…

We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its…

In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an…

We study the maximal ranks of a free and a free abelian quotients of a finitely generated group, called co-rank (inner rank, cut number) and the Betti number, respectively. We show that any combination of these values within obvious…

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups $G$: the degree of commutativity $\text{dc}_S(G)$, with…

On etudie divers aspects d'une formule qui compte les reflexions pleines dans les groupes de Coxeter finis. ***** We study several points about a formula which counts reflexions in a finite Coxeter group whose reduced decompositions involve…

The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free non-cyclic subgroup are proved.

We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the…

A $\lambda$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion…

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…

Using the classification of finite Weyl groupoids we prove that crystallographic arrangements, a large subclass of the class of simplicial arrangements which was recently defined, are hereditarily inductively free. In particular, all…

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are…

We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B<F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann.…

In this thesis, we study the combinatorics of cyclically fully commutative elements in Coxeter groups of type $A$ as it relates to conjugacy. In particular, we introduce the notion of cylindrical heaps and ring equivalence in order to state…