*(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 | Decision problems (group theory); connections with logic and automata |

20E45 | Conjugacy classes |

20F05 | Generators, relations, and presentations of groups |

68W30 | Symbolic computation and algebraic computation |