Conjugacy in Garside Groups III: Periodic braids

Details Conjugacy in Garside Groups III: Periodic braids Conjugacy in Garside Groups III: Periodic braids

2006-09-21

arxiv.org/abs/math/0609616v2

Mathematics

Geometric Topology

33 pages, 13 figures. Classical references implying Corollaries 12 and 15 have been added. To appear in Journal of Algebra

Scientific paper

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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