Related papers: Braid groups of J-reflection groups and associated…
We construct a family of links we call torus necklaces for which the link groups are precisely the braid groups of generalised $J$-reflection groups. Moreover, this correspondence exhibits the meridians of the aforementioned link groups as…
Complex reflection groups of rank two are precisely the finite groups in the family of groups that we call J-reflection groups. These groups are particular cases of J-groups as defined by Achar & Aubert in 2008. The family of J-reflection…
The family of J-groups was introduced by Achar and Aubert with the goal of providing Coxeter-like combinatorial tools for studying rank 2 complex reflection groups. However, J-groups lack an explicit presentation with abstract reflections…
Several finite complex reflection groups have a braid group which is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order $k$ for some $k\geq 2$, and meridians are…
We describe a new presentation for the complex reflection groups of type $(e,e,r)$ and their braid groups. A diagram for this presentation is proposed. The presentation is a monoid presentation which is shown to give rise to a Garside…
In his seminal paper on complex reflection arrangements, Bessis introduces a Garside structure for the braid group of a well-generated irreducible complex reflection group. Using this Garside structure, he establishes a strong connection…
After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural…
We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with…
Brou\'e, Malle and Rouquier conjectured in that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for…
Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer…
We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this…
We define geodesic normal forms for the general series of complex reflection groups G(e,e,n). This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of…
In this paper we introduce a class of `parabolic' subgroups for the generalized braid group associated to an arbitrary irreducible complex reflection group, which maps onto the collection of parabolic subgroups of the reflection group.…
The orbifold braid groups of two dimensional orbifolds were defined in [1] (arXiv:math/9907194) to understand certain Artin groups as subgroups of some suitable orbifold braid groups. We studied orbifold braid groups in some more detail in…
We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will…
We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its…
Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid…
A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations…
Define a Garside monoid to be a cancellative monoid where right and left lcm's exist and that satisfy additional finiteness assumptions, and a Garside group to be the group of fractions of a Garside monoid. The family of Garside groups…
Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e…