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**Conjugacy in Garside groups. I: Cyclings, powers and rigidity.**
*(English)*
Zbl 1160.20026

Summary: In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where ‘rigid’ means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given \(X\) in a Garside group, if some power \(X^m\) is conjugate to a rigid element, then \(m\) can be bounded above by \(\|\Delta\|^3\). In the particular case of braid groups \(\{B_n;\;n\in\mathbb{N}\}\), this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors’ program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.

### 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 |

### Keywords:

Garside groups; rigidity; stable ultra summit sets; iterated cyclings; iterated powers; normal form theorems; braid groups; conjugacy decision problem; conjugacy search problem
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\textit{J. S. Birman} et al., Groups Geom. Dyn. 1, No. 3, 221--279 (2007; Zbl 1160.20026)

### References:

[1] | S. I. Adyan, Fragments of the word ^ in the braid group. Mat. Zametki 36 (1984), 25-34; English transl. Math. Notes 36 (1984), 505-510. · Zbl 0599.20044 |

[2] | I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), 287-291. · Zbl 0944.94012 |

[3] | E. Artin, Theorie der Zöpfe. Abh. Math. Sem. Univ. Hamburg 4 (1925), 47-72. · JFM 51.0450.01 |

[4] | D. Benardete, M. Gutiérrez, and Z. Nitecki, A combinatorial approach to reducibility of mapping classes. In Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math. 150,Amer. Math. Soc., Providence, RI, 1993, 1-31. · Zbl 0804.57005 |

[5] | D. Bernardete, Z. Nitecki, and M. Gutiérrez, Braids and the Nielsen-Thurston classifica- tion. J. Knot Theory Ramifications 4 (1995), 549-618. · Zbl 0874.57010 |

[6] | D. Bessis, The dual braid monoid. Ann. Sci. École Norm. Sup. (4) 36 (2003), 647-683. · Zbl 1064.20039 |

[7] | D. Bessis, F. Digne, and J. Michel, Springer theory in braid groups and the Birman-Ko-Lee monoid. Pacific J. Math. 205 (2002), 287-309. · Zbl 1056.20023 |

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

[9] | J. S. Birman, V. Gebhardt, and J. González-Meneses, Conjugacy in Garside groups II: structure of the ultra summit set. Groups Geom. Dyn. , to appear; preprint 2006, |

[10] | J. S. Birman, V. Gebhardt, and J. González-Meneses, Conjugacy in Garside groups III: periodic braids. J. Algebra , to appear; preprint 2006, |

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

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