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Algorithmes et généricité dans les groupes de tresses

Abstract : The theory of braid groups is at the intersection of several areas of mathematics, especially algebra and geometry. The current research extends in each of these directions, leading to rich developments. From a geometrical point of view, the braid group on n strands is seen as the mapping class group of a disc with n punctures, with boundary component. A braid can be represented by a curve diagram, that is to say, the image of a family of arcs attached to the disc, by the corresponding mapping class. In this thesis we present the algorithm of relaxations from the right, which, given a curve diagram, determines the braid from which it was obtained. This algorithm helps us to make the link between geometric properties of the curve diagram and algebraic properties of the braid word, allowing us to identify great powers of a generator as spirals in the curve diagram. From an algebraic point of view, the braid group is the classical example of a Garside group. One of the objectives of current research in Garside theory is to obtain a polynomial time algorithm to solve the conjugacy problem in braid groups. For this, a possibility is to exploit the properties of some finite sets of conjugates of a braid, which are invariants of the conjugacy classes. One of the results of this thesis concerns the size of one of these invariants, the super summit set: we construct a family of pseudo-Anosov braids whose super summit set has exponential size. González- Meneses had already established the similar result for a family of reducible braids. These results implies that we cannot hope to solve the conjugacy problem in polynomial time through this set, and it is better to try to use smaller invariants. In the case of pseudo-Anosov braids, one may hope that the so-called sliding circuit set is more useful. In this thesis, we present a polynomial time algorithm based on this last set which generically solves the conjugacy problem, that is to say, it solves it for a proportion of braids that tends exponentially fast to 1 as the length of the braid tends to infinity. We also show that, in a ball of the Cayley graph with generators the simple braids, a braid is generically pseudo-Anosov, which was a well-known conjecture for the specialists in Garside theory.
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Submitted on : Friday, November 8, 2013 - 1:43:51 PM
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Sandrine Caruso. Algorithmes et généricité dans les groupes de tresses. Mathématiques générales [math.GM]. Université Rennes 1, 2013. Français. ⟨NNT : 2013REN1S058⟩. ⟨tel-00881511⟩



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