Conjugacy in Garside groups III: Periodic braids

By J. Birman, Volker Gebhardt, Juan González-Meneses. 2006

57 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d76c33f0357e563e380859f5a9090a8c195ccaa

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In Geometry and Topology

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

In Mathematical Physics

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

Citations in the corpus

• The cyclic sliding operation in Garside groups (2008, methodology): “It follows, as noticed in [7], that |USS(δ)| = 2n−2.”
• Conjugacy in Garside groups I: cyclings, powers and rigidity (2006, result): “The work in this paper and [9], [ 10 ] was done simultaneously and independently from the work of S. J. Lee and E. K. Lee in [30], [31], [32].”
• Basic results on braid groups (2010, background): “In the periodic case, there is a polynomial solution given in [11].”
• Foundations of Garside Theory (2013, unknown): “This program turned out to be rather successful and it led to many developments by a number of different authors [8, 11, 12, 16, 18, 17, 48, 47, 59, 111, 118, 127, 128, 158, 157, 165, 181, 192, ...].”
• Conjugacy in Garside groups II: structure of the ultra summit set (2006, methodology): “The focus of this paper and of its companions [3] and [4] is on gaining an improved understanding of the known algorithmic solutions in Garside groups of the conjugacy decision problem (CDP), that is, to decide, for arbitrary X; Y 2 G whether there exists Z such that Y D Z XZ; and also the conjugacy”