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Conjugacy in Garside groups. III: Periodic braids. (English) Zbl 1165.20031
Summary: 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 [for part II cf. Groups Geom. Dyn. 2, No. 1, 13-61 (2008; Zbl 1163.20023)]. 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.

##### MSC:
 20F36 Braid groups; Artin groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E45 Conjugacy classes for groups 20F05 Generators, relations, and presentations of groups 68W30 Symbolic computation and algebraic computation
Magma
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