Irreducible Coxeter groups

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 to commensurability and a permutation of the factors

From braid groups to mapping class groups

This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups

Artin groups of spherical type up to isomorphism

We prove that two Artin groups of spherical type are isomorphic if and only if their defining Coxeter graphs are the same

Mapping class groups of non-orientable surfaces for beginners

The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces

Birman's conjecture for singular braids on closed surfaces

Let $M$ be a closed oriented surface of genus $g\ge 1$, let $B_n(M)$ be the braid group of $M$ on $n$ strings, and let $SB_n(M)$ be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map $\eta: SB_n(M) \to \Z [B_n(M)]$, introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective

Commensurators of parabolic subgroups of Coxeter groups

Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup of $W$ generated by $X$ is denoted by $W_X$ and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of $W_X$ in $W$ is the subgroup of $w$ in $W$ such that $wW_Xw^{-1}\cap W_X$ has finite index in both $W_X$ and $wW_Xw^{-1}$. The subgroup $W_X$ can be decomposed in the form $W_X = W_{X^0} \cdot W_{X^\infty} \simeq W_{X^0} \times W_{X^\infty}$ where $W_{X^0}$ is finite and all the irreducible components of $W_{X^\infty}$" > are infinite. Let $Y^\infty$ be the set of $t$ in $S$ such that $m_{s,t}=2$" > for all $s\in X^\infty$. We prove that the commensurator of $W_X$ is $W_{Y^\infty} \cdot W_{X^\infty} \simeq W_{Y^\infty} \times W_{X^\infty}$. In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and $W_X$ is its own commensurator if and only if $X^0=Y^\infty$.Comment: Plain tex version, 9 pages no figure

The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group

It was conjectured by Tits that the only relations amongst the squares of the standard generators of an Artin group are the obvious ones, namely that a^2 and b^2 commute if ab=ba appears as one of the Artin relations. In this paper we prove Tits' conjecture for all Artin groups. More generally, we show that, given a number m(s)>1 for each Artin generator s, the only relations amongst the powers s^m(s) of the generators are that a^m(a) and b^m(b) commute if ab=ba appears amongst the Artin relations.Comment: 18 pages, 11 figures (.eps files generated by pstricks.tex). Prepublication du Laboratoire de Topologie UMR 5584 du CNRS (Univ. de Bourgogne

HOMFLY-PT skein module of singular links in the three-sphere

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links