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Zbl 1165.20031
Birman, Joan S.; Gebhardt, Volker; González-Meneses, Juan
Conjugacy in Garside groups. III: Periodic braids. (English)
[J] J. Algebra 316, No. 2, 746-776 (2007). ISSN 0021-8693

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.\par 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 2000:: *20F36 Braid groups; Artin groups
20F10 Decision problems (group theory)
20E45 Conjugacy classes
20F05 Presentations of groups
68W30 Symbolic computation and algebraic computation

Keywords: conjugacy problem; braid groups; periodic elements; Artin groups; algorithms; conjugacy search problem; conjugacy problem; Garside groups; conjugacy decision problem; Artin-Tits groups

Citations: Zbl 1163.20023

Cited in: Zbl 1230.20041

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