The cyclic sliding operation in Garside groups.

*(English)*Zbl 1253.20034From 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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##### References:

[1] | Adyan S.I.: Fragments of the word {\(\Delta\)} in the braid group (Russian). Mat. Zamet. 36, 25–34 (1984) · Zbl 0599.20044 |

[2] | Benardete D., Gutierrez M., Nitecki Z.: A combinatorial approach to reducibility of mapping classes. Contemp. Math. 150, 1–31 (1993) · Zbl 0804.57005 |

[3] | Benardete D., Gutierrez M., Nitecki Z.: Braids and the Nielsen–Thurston classification. J. Knot Theory Ramif. 4, 549–618 (1995) · Zbl 0874.57010 |

[4] | Bestvina M., Handel M.: Train-tracks for surface homeomorphisms. Topology 34, 109–140 (1995) · Zbl 0837.57010 |

[5] | Birman J.S., Gebhardt V., González-Meneses J.: Conjugacy in Garside groups I: cycling, powers and rigidity. Groups Geom. Dyn. 1, 221–279 (2007) · Zbl 1160.20026 |

[6] | Birman J.S., Gebhardt V., González-Meneses J.: Conjugacy in Garside groups II: structure of the ultra summit set. Groups Geom. Dyn. 2, 16–31 (2008) · Zbl 1163.20023 |

[7] | Birman J.S., Gebhardt V., González-Meneses J.: Conjugacy in Garside groups III: periodic braids. J. Algebra 316, 746–776 (2007) · Zbl 1165.20031 |

[8] | Birman J.S., Ko K.Y., Lee S.J.: A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139, 322–353 (1998) · Zbl 0937.20016 |

[9] | Birman J.S., Ko K.Y., Lee S.J.: The infimum, supremum and geodesic length of a braid conjugacy class. Adv. Math. 164, 41–56 (2001) · Zbl 1063.20039 |

[10] | Birman J.S., Lubotzky A., McCarthy J.: Abelian and solvable subgroups of the mapping class groups. Duke Math. 50, 1107–1120 (1983) · Zbl 0551.57004 |

[11] | Charney R.: Artin groups of finite type are biautomatic. Math. Ann. 292, 671–683 (1992) · Zbl 0782.57001 |

[12] | Dehornoy P., Paris L.: Gaussian groups and Garside groups, two generalizations of Artin groups. Proc. Lond. Math. Soc. 79, 569–604 (1999) · Zbl 1030.20021 |

[13] | Dehornoy P.: Groupes de Garside. Ann. Sci. Ec. Norm. Super. 35, 267–306 (2002) · Zbl 1017.20031 |

[14] | Deligne P.: Les immeubles des groupes de tresses généralisés. Invent. Math. 17, 273–302 (1972) · Zbl 0238.20034 |

[15] | ElRifai E., Morton H.: Algorithms for positive braids. Q. J. Math. Oxf. Ser. II 45, 479–497 (1994) · Zbl 0839.20051 |

[16] | Epstein D.B.A., Cannon J.W., Holt D.F., Levy S.V.F., Patterson M.S., Thurston W.P.: Word Processing in Groups. Jones and Bartlett, Boston (1992) |

[17] | Franco N., González-Meneses J.: Conjugacy problem for braid groups and Garside groups. J. Algebra 266, 112–132 (2003) · Zbl 1043.20019 |

[18] | Garside F.: The braid group and other groups. Q. J. Math. Oxf. Ser. II 20, 235–254 (1969) · Zbl 0194.03303 |

[19] | Gebhardt V.: A new approach to the conjugacy problem in Garside groups. J. Algebra 292, 282–302 (2005) · Zbl 1105.20032 |

[20] | Gebhardt, V., González-Meneses, J.: Solving the conjugacy problem in Garside groups by cyclic sliding. arXiv:0809.0948v1 (2008) · Zbl 1235.20032 |

[21] | González Manchón P.: There exist conjugate simple braids whose associated permutations are not strongly conjugate. Math. Proc. Camb. Philos. Soc. 143, 663–667 (2007) · Zbl 1128.20022 |

[22] | Lee E.-K., Lee S.J.: Abelian subgroups of Garside groups. Commun. Algebra 36, 1121–1139 (2008) · Zbl 1155.20037 |

[23] | Lee E.-K., Lee S.J.: A Garside-theoretic approach to the reducibility problem in braid groups. J. Algebra 320, 783–820 (2008) · Zbl 1191.20034 |

[24] | Lee E.-K., Lee S.J.: Some power of an element in a Garside group is conjugate to a periodically geodesic element. Bull. Lond. Math. Soc. 40, 593–603 (2008) · Zbl 1171.20026 |

[25] | Lee, S.J.: Algorithmic solutions to decision problems in the braid groups. PhD thesis, Korea Advanced Institute of Science and Technology (2000) |

[26] | Zheng, H.: General cycling operations in Garside groups. arXiv:math/0605741v2 (2006) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.