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Conjugacy in Garside Groups III: Periodic braids

Journal of Algebra (Impact Factor: 0.6). 10/2006; 316(2). DOI: 10.1016/j.jalgebra.2007.02.002
Source: arXiv

ABSTRACT An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.

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    • "In the pseudo-Anosov case, they also give the structure of the unstable foliation F u , its measure and the dilatation factor λ. The dynamic of a braid only depends on its conjugacy class, in particular so does its Nielsen-Thurston type. So we can study the geometry of any given β ∈ B n , through any conjugate β = α −1 βα: Periodicity is easily recognizable in braid groups [5], and β k = ∆ 2s if and only if β k = ∆ 2s . If β is reducible, and C is a reduction system for β (1-manifold such that β(C) = C), then α(C) is a reduction system for β. "
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    • "If they are not the same, answer " x and y are not conjugate " and STOP. (2) If x and y are periodic use [4] and STOP. (3) If x and y are reducible, use [9] and STOP. "
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    • "As a self-homeomorphism on n-punctured 2-disk, an n-braid β is of one of the following three dynamic types known as Nielsen-Thurston classification [19]: (1) β is periodic if some power of β is isotopic to some power of Dehn twist along the boundary of disk; (2) β is reducible if there exists a family of pairwise disjoint essential non-peripheral simple closed curves in D n , called reduction system, which is preserved by a n-braid isotopic to β; (3) β is pseudo-Anosov is neither (1) nor (2). The conjugacy problem for periodic braids is relatively easy [11] [4]. For reducible braids, once the reduction system is known, the conjugacy problem can be reduced into several pieces, which are for either periodic or pseudo-Anosov braids. "
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