20 research outputs found
Cohomology of Artin groups of type tilde{A}_n, B_n and applications
We consider two natural embeddings between Artin groups: the group
G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type
B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of
type A_n. The cohomologies of these groups are related, by standard results, in
a precise way. By using techniques developed in previous papers, we give
precise formulas (sketching the proofs) for the cohomology of G_{B_n} with
coefficients over the module Q[q^{+-1},t^{+-1}], where the action is
(-q)-multiplication for the standard generators associated to the first n-1
nodes of the Dynkin diagram, while is (-t)-multiplication for the generator
associated to the last node.
As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as well
as the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensional
representation obtained by Tong, Yang and Ma.
We stress the topological significance, recalling some constructions of
explicit finite CW-complexes for orbit spaces of Artin groups. In case of
groups of infinite type, we indicate the (few) variations to be done with
respect to the finite type case. For affine groups, some of these orbit spaces
are known to be K(pi,1) spaces (in particular, for type tilde{A}_n).
We point out that the above cohomology of G_{B_n} gives (as a module over the
monodromy operator) the rational cohomology of the fibre (analog to a Milnor
fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.Comment: This is the version published by Geometry & Topology Monographs on 22
February 200
T-structures on some local Calabi-Yau varieties
Let be a Fano variety satisfying the condition that the rank of the
Grothendieck group of is one more than the dimension of . Let
denote the total space of the canonical line bundle of , considered as a
non-compact Calabi-Yau variety. We use the theory of exceptional collections to
describe t-structures on the derived category of coherent sheaves on
. The combinatorics of these t-structures is determined by a natural
action of an affine braid group, closely related to the well-known action of
the Artin braid group on the set of exceptional collections on .Comment: 30 page
Braid groups of imprimitive complex reflection groups
We obtain new presentations for the imprimitive complex reflection groups of
type and their braid groups for . Diagrams
for these presentations are proposed. The presentations have much in common
with Coxeter presentations of real reflection groups. They are positive and
homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms
correspond to group automorphisms. The new presentation shows how the braid
group is a semidirect product of the braid group of affine type
and an infinite cyclic group. Elements of are
visualized as geometric braids on strings whose first string is pure and
whose winding number is a multiple of . We classify periodic elements, and
show that the roots are unique up to conjugacy and that the braid group
is strongly translation discrete.Comment: published versio
and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups
Let be a Coxeter graph, let be its associated Coxeter
system, and let ) be its associated Artin-Tits system. We regard
as a reflection group acting on a real vector space . Let be the Tits
cone, and let be the complement in of the reflecting
hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a
simplicial complex having the same homotopy type as
. We observe that, if , then
naturally embeds into . We prove that this embedding admits a
retraction , and we deduce several
topological and combinatorial results on parabolic subgroups of . From a
family \SS of subsets of having certain properties, we construct a cube
complex , we show that has the same homotopy type as the universal
cover of , and we prove that is CAT(0) if and only if \SS is
a flag complex. We say that is free of infinity if has
no edge labeled by . We show that, if is aspherical and
has a solution to the word problem for all free of
infinity, then is aspherical and has a solution to the word
problem. We apply these results to the virtual braid group . In
particular, we give a solution to the word problem in , and we prove that
the virtual cohomological dimension of is