531 research outputs found

    Braid groups of imprimitive complex reflection groups

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    We obtain new presentations for the imprimitive complex reflection groups of type (de,e,r)(de,e,r) and their braid groups B(de,e,r)B(de,e,r) for d,r≥2d,r \ge 2. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group B(de,e,r)B(de,e,r) is a semidirect product of the braid group of affine type A~r−1\widetilde A_{r-1} and an infinite cyclic group. Elements of B(de,e,r)B(de,e,r) are visualized as geometric braids on r+1r+1 strings whose first string is pure and whose winding number is a multiple of ee. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group B(de,e,r)B(de,e,r) is strongly translation discrete.Comment: published versio

    Periodic elements in Garside groups

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    Let GG be a Garside group with Garside element Δ\Delta, and let Δm\Delta^m be the minimal positive central power of Δ\Delta. An element g∈Gg\in G is said to be 'periodic' if some power of it is a power of Δ\Delta. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of GG is cyclic; if gk=Δkag^k=\Delta^{ka} for some nonzero integer kk, then gg is conjugate to Δa\Delta^a; every finite subgroup of the quotient group G/G/ is cyclic. By a classical theorem of Brouwer, Ker\'ekj\'art\'o and Eilenberg, an nn-braid is periodic if and only if it is conjugate to a power of one of two specific roots of Δ2\Delta^2. We generalize this to Garside groups by showing that every periodic element is conjugate to a power of a root of Δm\Delta^m. We introduce the notions of slimness and precentrality for periodic elements, and show that the super summit set of a slim, precentral periodic element is closed under any partial cycling. For the conjugacy problem, we may assume the slimness without loss of generality. For the Artin groups of type AnA_n, BnB_n, DnD_n, I2(e)I_2(e) and the braid group of the complex reflection group of type (e,e,n)(e,e,n), endowed with the dual Garside structure, we may further assume the precentrality.Comment: The contents of the 8-page paper "Notes on periodic elements of Garside groups" (arXiv:0808.0308) have been subsumed into this version. 27 page

    The dual braid monoid

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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-LeeComment: 42 pages. Major revision, many new result

    Braids: A Survey

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    This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labelled, as we encounter them.Comment: Final version, revised to take account of the comments of readers. A review article, to appear in the Handbook of Knot Theory, edited by W. Menasco and M. Thistlethwaite. 91 pages, 24 figure

    On the cycling operation in braid groups

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    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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