Search results for "20F36"
showing 10 items of 25 documents
Commensurability in Artin groups of spherical type
2019
Let $A$ and $A'$ be two Artin groups of spherical type, and let $A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$ (resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A'_i$ are commensurable for every $i$. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed $n$, we give a complete classification of the irreducible Artin groups of rank $n$ that are commensurable with the group of type $A_n$. Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensur…
PreGarside monoids and groups, parabolicity, amalgamation, and FC property
2012
We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin–Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parabolic subgroups in terms of these presentations. Afterwards we prove that the amalgamated product of two preGarside groups along a common parabolic subgroup is again a preGarside group. This enables us to define the family of preGarside groups of FC type as the smallest family of preGarside groups that contains the Garside g…
Artin groups of spherical type up to isomorphism
2003
AbstractWe prove that two Artin groups of spherical type are isomorphic if and only if their defining Coxeter graphs are the same.
Vassiliev invariants for braids on surfaces
2000
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.
Automorphisms and abstract commensurators of 2-dimensional Artin groups
2004
In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism group and generated by inner automorphisms, graph automorphisms (induced from automorphisms of the…
Categorical action of the extended braid group of affine type $A$
2017
Using a quiver algebra of a cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type A on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.
Geometric représentations of the braid groups
2010
We show that the morphisms from the braid group with n strands in the mapping class group of a surface with a possible non empty boundary, assuming that its genus is smaller or equal to n/2 are either cyclic morphisms (their images are cyclic groups), or transvections of monodromy morphisms (up to multiplication by an element in the centralizer of the image, the image of a standard generator of the braid group is a Dehn twist, and the images of two consecutive standard generators are two Dehn twists along two curves intersecting in one point). As a corollary, we determine the endomorphisms, the injective endomorphisms, the automorphisms and the outer automorphism group of the following grou…
A simple algorithm for finding short sigma-definite representatives
2010
We describe a new algorithm which for each braid returns a quasi-geodesic sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears either only positively or only negatively.
Les tresses : de la topologie à la cryptographie
2009
International audience; Ce texte est une présentation sur les groupes de tresses destinée à un public de non mathématiciens.
Quasi-isometrically embedded subgroups of braid and diffeomorphism groups
2005
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F\_n$ and $\Z^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the diffeomorphism group of the disk. Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundame…