Publications et prépublications
en liaison avec les tresses
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| Abstract:
We describe new combinatorial methods for constructing an explicit free resolution of~$\ZZ$ by $\ZZ G$-modules when $G$ is a group of fractions
of a monoid where enough least common multiples exist (``locally Gaussian monoid''), and, therefore, for computing the homology of~$G$. Our
constructions apply in particular to all Artin groups of finite Coxeter type, so, as a corollary, they give new ways of computing the homology of these
groups.
MSC : 20J06, 18G35, 20M50, 20F36. |
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On construit une classe de morphismes entre groupes d'Artin-Tits qui généralise celle construite par J. Crisp. Nous montrons que leurs restrictions
aux monoïdes respectent les formes normales, et que pour les groupes d'Artin-Tits de type FC ces morphismes sont injectifs. La démonstration du
second résultat utilise le complexe de Deligne et les chemins cubiques normaux.
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Séminaire Lotharingien de Combinatoire, 1997, vol. 37, 36 pages. MSC : |
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Rapport interne du LaBRI no. 1219-99, Mai 1999, 25 pages. MSC : |
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Conférence invitée, Dans ``Formal Power Series and Algebraic Combinatorics, 12th International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings'' (D. Krob, A. A. Mikhalev, A. V. Mikhalev, eds.), Springer, 2000, 76--87. MSC : |
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A paraître dans "The Moscow Mathematical Journal" (34 pages). MSC : |
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Nous établissons, d'une part, d'une manière théorique (via l'étude d'une représentation linéaire du groupe de tresse) et, d'autre part, d'une manière
algorithmique, une correspondance entre deux types de description des variétés topologiques de dimension trois. Plus précisement, si $p : M -> S3$
est un revêtement simple trois feuillets de la sphère ramifié le long d'un entrelacs, nous donnons une façon de trouver un entrelacs $L$ de $S3$ et des
coefficients de chirurgie tels que $M$ soit homéomorphe au resultat de la chirurgie sur $S3$ le long de $L$.
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Let $(A,S)$ be an Artin-Tits group and $T$ a subset of $S$. We denote by $A_T$ the subgroup of $A$ generated by $T$ and $A^+$, $A^+_T$ the
sub-monoids of $A$ and $A_T$ generated by $S$ and $T$ respectively. We prove that the quasi-centralizer of $A^+_T$ in $A^+$ consists of
``ribbons''; this notion generalizing the notion of ribbon due to Fenn, Rolfsen and Zhu and that of conjugator due to Paris. Furthermore we give, under
an additional hypothesis, a finite set of positive generators of the centralizer and of the quasi-centralizer of $A_T$ in $A$. Finally we study the
normalizer and commensurator of $A_T$ in $A$ and prove that when $A$ is spherical, they are equal and are the product of $A_T$ by the
quasi-centralizer; in that way this generalizes a result of Paris.
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We will prove that if an element of an Artin monoid is conjugated to a power $s^j$ of a generator $s$ in the corresponding Artin group then it is equal
to the power $t^j$ of another generator $t$. Furthermore the centraliser in the monoid of a power of a generator is equal to the centraliser of that
generator.
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Let $(A,S)$ be an Artin group of type FC and $A_T$ a standard parabolic subgroup of $A$. We use combinatorial tools to show that the normalizer of
$A_T$,the commensurator of $A_T$, and the product of thequasi-centralizer of $A_T$ by $A_T$ are egal. Furthermore, we show that the centralizer
and the quasi-centralizer of $A_T$ in $A$ are generated by their intersections with the monoid $A^+$.
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A combinatorial property of (semi)group presentations, called completeness, is introduced. We give an effective criterion for recognizing finite
complete presentations, and for possibly completing an incomplete presentation. We show how several properties, like cancellativity or existence of
common multiples in the associated monoid, or isoperimetric inequality in the associated group can be easily decided when a (finite) complete
presentation is available. In particular, we obtain a new criterion for proving that a monoid embeds in a group of fractions. Typical presentations
eligible for the current approach are the standard presentations of Artin groups, but also quite different ones such as the standard presentation of the
Heisenberg group.
MSC : 20M05, 05C25, 68Q42, 20F36. |
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A number of properties of spherical Artin groups extend to thin Gaussian groups, the latter being defined as the groups of fractions of monoids where
least common multiples exist and there is no nontrivial unit. Here we investigate a wider class of groups of fractions, called thin, which are associated
with monoids where minimal common multiples exist, but they are not necessarily unique. Also, we allow units in the involved monoids. The main
results are that all thin groups of fractions satisfy a quadratic isoperimetric inequality, and that, under some additional hypotheses, they admit an
automatic structure.
MSC : 20M05, 05C25, 68Q42, 20F36. |
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J. Phys. (A): Math. Gen., 29 (1996), 2411-2434 MSC : |
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Thin Gaussian groups are a natural generalization of spherical Artin groups, namely groups of fractions of monoids in which the existence of least
common multiples is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be of Coxeter type. Here we
completely describe the center of thin Gaussian groups by constructing a minimal generating set for the quasi-center. We deduce that every thin
Gaussian group is an iterated crossed product of thin Gaussian groups with a cyclic center.
J. of Algebra, 245-1 (2001) 92--122. MSC : 20F05, 20F36, 20F12. |
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We construct a new monoid structure for Artin groups associated with finite Coxeter systems. This monoid shares with the classical positive braid
monoid a crucial algebraic property: it is a Garside monoid. The analogy with the classical construction indicates there is a ``dual'' way of studying
Coxeter systems, where the pair (W,S) is replaced by (W,T), with T the set of all reflections. In the type A case, we recover the monoid constructed by
Birman-Ko-Lee.
MSC : 20F36; 20F55. |
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Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and
Broué-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for presentations of braid groups, which
partially explains and generalizes the known empirical properties. Our approach is invariant-theoretic and does not use the classification. The two
ingredients are Springer theory of regular elements and a Zariski-like theorem.
MSC : 20F36. |
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V. B. Styshnev showed that the existence of $n$-th roots for a braid is decidable. Thin Gaussian groups have been introduced by P.Dehornoy as a
natural proper generalization of Artin groups of finite type. We have to construct a new proof to extend Styshnev's decidability result to thin Gaussian
groups, as several specific properties of braids that were used fail in our case. We show that, under the assumption of a finiteness property of
conjugacy, the problem is decidable.
MSC : 20F36, 20B40, 20E45, 20F05. |
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We prove that the natural homomorphism from an Artin monoid to its associated Artin group is always injective.
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