171 research outputs found
Parabolic subgroups of Garside groups II: ribbons
We introduce and investigate the ribbon groupoid associated with a Garside
group. Under a technical hypothesis, we prove that this category is a Garside
groupoid. We decompose this groupoid into a semi-direct product of two of its
parabolic subgroupoids and provide a groupoid presentation. In order to
established the latter result, we describe quasi-centralizers in Garside
groups. All results hold in the particular case of Artin-Tits groups of
spherical type
Generic Hecke algebra for Renner monoids
We associate with every Renner monoid a \emph{generic Hecke algebra}
over which is a deformation of the monoid
-algebra of . If is a finite reductive monoid with Borel
subgroup and associated Renner monoid , then we obtain the associated
Iwahori-Hecke algebra by specialising in and tensoring by
over , as in the classical case of finite algebraic
groups. This answers positively to a long-standing question of L. Solomon
Basic Questions on Artin-Tits groups
This paper is a short survey on four basic questions on Artin-Tits groups:
the torsion, the center, the word problem, and the cohomology (
problem). It is also an opportunity to prove three new results concerning these
questions: (1) if all free of infinity Artin-Tits groups are torsion free, then
all Artin-Tits groups will be torsion free; (2) If all free of infinity
irreducible non-spherical type Artin-Tits groups have a trivial center then all
irreducible non-spherical type Artin-Tits groups will have a trivial center;
(3) if all free of infinity Artin-Tits groups have solutions to the word
problem, then all Artin-Tits groups will have solutions to the word problem.
Recall that an Artin-Tits group is free of infinity if its Coxeter graph has no
edge labeled by
A conjecture about Artin-Tits groups
We conjecture that the word problem of Artin-Tits groups can be solved
without introducing trivial factors ss^{-1} or s^{-1}s. Here we make this
statement precise and explain how it can be seen as a weak form of
hyperbolicity. We prove the conjecture in the case of Artin-Tits groups of type
FC, and we discuss various possible approaches for further extensions, in
particular a syntactic argument that works at least in the right-angled case
Questions on surface braid groups
We provide new group presentations for surface braid groups which are
positive. We study some properties of such presentations and we solve the
conjugacy problem in a particular case
Folding of set-theoretical solutions of the Yang-Baxter equation
We establish a correspondence between the invariant subsets of a
non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter
equation and the parabolic subgroups of its structure group, equipped with its
canonical Garside structure. Moreover, we introduce the notion of a foldable
solution, which extends the one of a decomposable solution
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
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