259 research outputs found
The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group
It was conjectured by Tits that the only relations amongst the squares of the
standard generators of an Artin group are the obvious ones, namely that a^2 and
b^2 commute if ab=ba appears as one of the Artin relations. In this paper we
prove Tits' conjecture for all Artin groups. More generally, we show that,
given a number m(s)>1 for each Artin generator s, the only relations amongst
the powers s^m(s) of the generators are that a^m(a) and b^m(b) commute if ab=ba
appears amongst the Artin relations.Comment: 18 pages, 11 figures (.eps files generated by pstricks.tex).
Prepublication du Laboratoire de Topologie UMR 5584 du CNRS (Univ. de
Bourgogne
Multifraction reduction III: The case of interval monoids
We investigate gcd-monoids, which are cancellative monoids in which any two
elements admit a left and a right gcd, and the associated reduction of
multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the
word problem for the enveloping group. Here we consider the particular case of
interval monoids associated with finite posets. In this way, we construct
gcd-monoids, in which reduction of multifractions has prescribed properties not
yet known to be compatible: semi-convergence of reduction without convergence,
semi-convergence up to some level but not beyond, non-embeddability into the
enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added,
typos corrected; final version due to appear in Journal of Combinatorial
Algebr
Multifraction reduction II: Conjectures for Artin-Tits groups
Multifraction reduction is a new approach to the word problem for Artin-Tits
groups and, more generally, for the enveloping group of a monoid in which any
two elements admit a greatest common divisor. This approach is based on a
rewrite system ("reduction") that extends free group reduction. In this paper,
we show that assuming that reduction satisfies a weak form of convergence
called semi-convergence is sufficient for solving the word problem for the
enveloping group, and we connect semi-convergence with other conditions
involving reduction. We conjecture that these properties are valid for all
Artin-Tits monoids, and provide partial results and numerical evidence
supporting such conjectures.Comment: 41 pages , v2 : cross-references updated , v3 : exposition improved,
typos corrected, final version due tu appear in Journal of Combinatorial
Algebr
A Deligne complex for Artin monoids
In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(\pi,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph
On the cycling operation in braid groups
The cycling operation is a special kind of conjugation that can be applied to
elements in Artin's braid groups, in order to reduce their length. It is a key
ingredient of the usual solutions to the conjugacy problem in braid groups. In
their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it
cycling problem} as a hard problem in braid groups that could be interesting
for cryptography. In this paper we give a polynomial solution to that problem,
mainly by showing that cycling is surjective, and using a result by Maffre
which shows that pre-images under cycling can be computed fast. This result
also holds in every Artin-Tits group of spherical type.
On the other hand, the conjugacy search problem in braid groups is usually
solved by computing some finite sets called (left) ultra summit sets
(left-USS), using left normal forms of braids. But one can equally use right
normal forms and compute right-USS's. Hard instances of the conjugacy search
problem correspond to elements having big (left and right) USS's. One may think
that even if some element has a big left-USS, it could possibly have a small
right-USS. We show that this is not the case in the important particular case
of rigid braids. More precisely, we show that the left-USS and the right-USS of
a given rigid braid determine isomorphic graphs, with the arrows reversed, the
isomorphism being defined using iterated cycling. We conjecture that the same
is true for every element, not necessarily rigid, in braid groups and
Artin-Tits groups of spherical type.Comment: 20 page
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