It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups.

Artin groups of finite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of finite type, we construct a space (simplicial complex) analogous to Teichmüller space that satisfies a weak nonpositive curvature condition and also a space ”at infinity ” analogous to the space of projective measured laminations. Using these constructs, we deduce several group-theoretic properties of Artin groups of finite type that are well-known in the case of braid groups. 1

Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples

Introduction Cet article est une 'etape vers la compr'ehension de la forme particuli`ere que prennent, dans le cas des groupes r'eductifs finis G F , les conjectures g'en'erales 'enonc'ees dans [Br] pour les blocs `a groupes de d'efaut ab'eliens des groupes finis "abstraits". Apr`es avoir v'erifi'e l'aspect de ces conjectures portant sur les caract`eres des groupes r'eductifs finis (cf. [BrMaMi]), il reste `a 'etablir la version, beaucoup plus profonde et difficile, portant sur les 'equivalences de cat'egories d'eriv'ees concern'ees. Certaines de ces 'equivalences devraient en particulier etre fournies par les cohomologies 'etales de faisceaux convenables sur les vari'et'es de Deligne-Lusztig Xw associ'ees `a certains 'el'ements r'eguliers w du groupe de Weyl W de G (cf. [BrMa], x1). Ceci implique des propri'et'es particuli`eres de la cohomologie 'et

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G�K � is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1- or 2-fellow traveler property. © 2000 Academic Press 1.

Recent work of Gromov, Epstein, Cannon, Thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups. (See [16],[17] and [14].) Many of these structures are preserved by taking graph products.

We dene the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A n , B n = C n and D n and the ane diagrams ~ A n , ~ B n , ~ C n and ~ D n as subgroups of the braid groups of various simple orbifolds. The cases D n , ~ B n , ~ C n and ~ D n are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except ~ A n the quotient is nite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type D n . AMS Classication: 20F36 Keywords: braid group, Artin group, orbifold, Garside element 1 1

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.

Abstract. Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey article

this paper on two families of arrangements of hyperplanes, to the fundamental group of which many well-known results on the pure braid group can be extended. Both of them, of course, contain the braid arrangement. These families are the "simplicial arrangements" and the "supersolvable arrangements". Note that there is another wellunderstood family of arrangements, the "reflection arrangements" (see [OT, Ch. 6] and [BMR]), which contains the braid arrangement, and which is not treated in the present paper.