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143
A new approach to the conjugacy problem in Garside groups
, 2008
"... The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practica ..."
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Cited by 67 (6 self)
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The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic approach to the conjugacy search problem in Garside groups. The results are likely to have implications for the security of recently proposed cryptosystems based on the hardness of problems related to the conjugacy (search) problem in braid groups.
Conjugacy problem for braid groups and Garside groups
, 2002
"... We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee [3]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among oth ..."
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Cited by 65 (8 self)
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We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee [3]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).
Springer Theory in Braid Groups and the BirmanKoLee Monoid
, 2000
"... We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for Artin braid groups and some other cases. Our proof makes use of results from BirmanKoLee, of which we give a new intrinsic account. ..."
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Cited by 60 (10 self)
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We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for Artin braid groups and some other cases. Our proof makes use of results from BirmanKoLee, of which we give a new intrinsic account.
The dual braid monoid
 Ann. Sci. (2003), 647–683. École Norm. Sup
"... Abstract. We describe a new monoid structure for braid groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid its crucial algebraic properties: the monoid satisfies Öre’s conditions and embeds in its group of fractions, it admits a nice normal form ..."
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Cited by 57 (2 self)
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Abstract. We describe a new monoid structure for braid groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid its crucial algebraic properties: the monoid satisfies Öre’s conditions and embeds in its group of fractions, it admits a nice normal form, it can be used to construct braid group actions on categories... It also provides a new presentation for braid groups; the conjugation by a Coxeter element is a “diagram automorphism ” of the new presentation. In the type A case, one recovers the BirmanKoLee presentation.
Complete positive group presentations
 Preprint; ArXiv math.GR/0111275. GROUPS OF FRACTIONS 33
"... Abstract. A combinatorial property of prositive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show how to directly read several properties of the assoc ..."
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Cited by 31 (19 self)
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Abstract. A combinatorial property of prositive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show how to directly read several properties of the associated monoid and group from a complete presentation: cancellativity or existence of common multiples in the case of the monoid, or isoperimetric inequality in the case of the group. In particular, we obtain a new criterion for recognizing that a monoid embeds in a group of fractions. Typical presentations eligible for the current approach are the standard presentations of the Artin groups and the Heisenberg group.
The Conjugacy Problem In Small Gaussian Groups
 Comm. in Algebra
, 2001
"... Small Gaussian groups are a natural generalization of spherical Artin groups 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. We show here how to extend the ElrifaiMorton solution ..."
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Cited by 30 (5 self)
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Small Gaussian groups are a natural generalization of spherical Artin groups 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. We show here how to extend the ElrifaiMorton solution for the conjugacy problem in braid groups to every small Gaussian group. Key words: conjugacy; word problem; Artin groups. MSC 2000: 20F05, 20F36, 20F12.
Conjugacy in Garside groups III: Periodic braids
 J. Algebra
"... An element in Artin’s braid group Bn is said to be periodic if some power of it lies in the center of Bn. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in Bn are exponential in the braid index n for the special case of periodic braids. We overco ..."
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Cited by 21 (5 self)
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An element in Artin’s braid group Bn is said to be periodic if some power of it lies in the center of Bn. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in Bn 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 Bn 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 Bn, 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 ArtinTits groups of spherical type. 1
Gaussian groups are torsion free
 J. Algebra
, 1998
"... Abstract. Assume that G is a group of fractions of a cancellative monoid where lower common multiples exist and divisibility has no infinite descending chain. Then G is torsion free. The result applies in particular to all finite Coxeter type Artin groups. Finding an elementary proof for the fact th ..."
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Cited by 20 (3 self)
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Abstract. Assume that G is a group of fractions of a cancellative monoid where lower common multiples exist and divisibility has no infinite descending chain. Then G is torsion free. The result applies in particular to all finite Coxeter type Artin groups. Finding an elementary proof for the fact that Artin’s braid groups are torsion free has been reported to be a longstanding open question [9]. The existence of a linear ordering of the braids that is left compatible with product [4] has provided such a proof—see also [10]. The argument applies to Artin groups of type Bn as well, but it remains rather specific, and there seems to be little hope to extend it to a much larger family of groups. On the other hand, we have observed in [5] and [6] that Garside’s analysis of the braids [8] applies to a large family of groups, namely all groups of fractions associated with certain monoids where divisibility has a lattice structure or, equivalently, all groups that admit a presentation of a certain syntactic form. Such groups have been called Gaussian in [6]. It is shown in the latter paper that all finite Coxeter