arXiv subjects
Publications and source records attributed to Luis Paris.
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 $η: SB_n(M) \to \Z [B_n(M)]$, introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.
From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $ρ: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation shall be called the Artin type representation associated to the pair $(H,h)$. The goal of the present paper is to study different aspects of these representations. Firstly, we associate to each braid $β$ a group $Γ_{(H,h)} (β)$ and prove that the operator $Γ_{(H,h)}$ determines a group invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant $Γ_{(H,h)}$, and we prove that the Artin type representations are faithful. The last part of the paper is dedicated to the study of some semidirect products $G \rtimes_ρB_n$, where $ρ: B_n \to \Aut(G)$ is an Artin type representation. In particular, we show that $G \rtimes_ρB_n$ is a Garside group if $H$ is a Garside group and $h$ is a Garside element of $H$.
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this monodromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot be the whole representation.
We determine the epimorphisms $A \to W$ from the Artin group $A$ of type $Γ$ onto the Coxeter group $W$ of type $Γ$, in case $Γ$ is an irreducible Coxeter graph of spherical type, and we prove that the kernel of the standard epimorphism is a characteristic subgroup of $A$. This generalizes an over 50 years old result of Artin.
Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix the boundary of F and such that h(P) = P. In this paper, we calculate presentations for all punctured mapping class groups. More precisely, we show that these groups are isomorphic with quotients of Artin groups by some relations involving fundamental elements of parabolic subgroups.
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.
We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered surface.
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 $X^0=Y^\infty$.