Summary: An element in Artin’s braid group is said to be periodic if some power of it lies in the center of . In this paper we prove that all previously known algorithms for solving the conjugacy search problem in are exponential in the braid index 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 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 , 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.