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**The cyclic sliding operation in Garside groups.**
*(English)*
Zbl 1253.20034

From the introduction: We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are rigid (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.

The structure of the paper is as follows. In Sect. 2 we give a basic introduction to the theory of Garside groups; specialists may skip this part, although the definition of local sliding in Sect. 2.2 should not be missed. In Sect. 3 we present the new concepts introduced in this paper: cyclic sliding in Sect. 3.1, the sets of sliding circuits in Sect. 3.2, the transport map in Sect. 3.3 and the sliding circuits graph in Sect. 3.4. Section 4 is devoted to theoretical applications of these new concepts: an algorithm to solve the conjugacy problem in Garside groups is explained in Sect. 4.1, applications to rigid elements – in particular the proofs of Theorems 1 and 2 – are given in Sect. 4.2, and finally we show in Sect. 4.3 that, in the particular case of the braid groups, the results which usually consider ultra summit sets to study reducible braids can also be translated to this new setting. Finally, Sect. 5 gives theoretical and computational examples comparing ultra summit sets to sets of sliding circuits in the case of braid groups.

The structure of the paper is as follows. In Sect. 2 we give a basic introduction to the theory of Garside groups; specialists may skip this part, although the definition of local sliding in Sect. 2.2 should not be missed. In Sect. 3 we present the new concepts introduced in this paper: cyclic sliding in Sect. 3.1, the sets of sliding circuits in Sect. 3.2, the transport map in Sect. 3.3 and the sliding circuits graph in Sect. 3.4. Section 4 is devoted to theoretical applications of these new concepts: an algorithm to solve the conjugacy problem in Garside groups is explained in Sect. 4.1, applications to rigid elements – in particular the proofs of Theorems 1 and 2 – are given in Sect. 4.2, and finally we show in Sect. 4.3 that, in the particular case of the braid groups, the results which usually consider ultra summit sets to study reducible braids can also be translated to this new setting. Finally, Sect. 5 gives theoretical and computational examples comparing ultra summit sets to sets of sliding circuits in the case of braid groups.

### 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 |

### Keywords:

Garside groups; cyclic sliding; conjugacy problem; algorithms; conjugacy search problem; cycling; ultra summit sets; decycling; super summit elements; rigid elements; braid groups
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\textit{V. Gebhardt} and \textit{J. González-Meneses}, Math. Z. 265, No. 1, 85--114 (2010; Zbl 1253.20034)

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