Related papers: Coxeter elements of the symmetric groups whose pow…
Two partial orders on a reflection group, the codimension order and the prefix order, are together called the absolute order when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner…
We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}.…
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,…
In this paper, we study the twist-conjecture for Coxeter systems and rigidity of Coxeter systems up to finite twists. For Coxeter systems $(W,R)$ and $(W,S)$, under the untangle-condition for conjugate subsets, we investigate separations…
We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to…
We study coherence of graph products and Coxeter groups and obtain many results in this direction.
For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and…
We refine the infinitesimal Hecke algebra associated to a 2-reflection group into a $\Z/2\Z$-graded Lie algebra, as a first step towards a global understanding of a natural $\mathbbm{N}$-graded object. We provide an interpretation of this…
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…
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…
We prove that |A^n| > c_n |A|^{[\frac{n+1}{2}]} for any finite subset A of a free group if A contains at least two noncommuting elements, where c_n>0 are constants not depending on A. Simple examples show that the order of these estimates…
A monoid $M$ generated by a set $S$ of symbols can be described as the set of equivalence classes of finite words in $S$ under some relations that specify when some contiguous sequence of symbols can be replaced by another. If $a,b\in S$, a…
In this short note, we provide an inequality that holds in any finite group, only involving the orders of the elements; we prove that equality holds if and only if the group is nilpotent.
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
An element of a Coxeter group $W$ is called fully commutative if any two of its reduced decompositions can be related by a series of transpositions of adjacent commuting generators. In the preprint "Fully commutative elements in finite and…
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined…
Folding subgroups give a way to realize non-simply-laced Coxeter groups as subgroups of simply-laced Coxeter groups. In this paper, we study how folding subgroups of finite and affine type are distributed length-wise by calculating the…
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…
A classical result of H. S. M. Coxeter asserts that a certain quotient $B(m,n)$ of the braid group $B(m)$ on $m$ strands is finite if and only if $(m,n)$ corresponds to the type of one of the five Platonic solids. If ${\bf k}$ is a knot or…
Let $\mathcal{W}$ be the set of strongly real elements of $W$, a Coxeter group. Then for $w \in \mathcal{W}$, $e(w)$, the excess of $w$, is defined by $e(w) = \min\{\ell(x) + \ell(y) - \ell(w) \; | \; w=xy, x^2 = y^2 = 1\}$. When $W$ is…