Conjugacy in Garside groups I: cyclings, powers and rigidity

By J. Birman, Volker Gebhardt, Juan González-Meneses. 2006

82 citations recorded by the source corpus.

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 kk 3 . In the particular case of braid groups fBnI n 2 Ng, 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/da3fb13e0bcf8e31914e58f8f1c59a487cd7fdc4

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In Discrete Mathematics and Combinatorics

1. Juan González-Meneses
2. J. Birman
3. Volker Gebhardt

In Geometry and Topology

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

In Mathematical Physics

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

Citations in the corpus

• The cyclic sliding operation in Garside groups (2008, unknown): “If one is able to detect efficiently the canonical reduction system of a braid, the CDP/CSP can be split into simpler problems, as explained in [5].”
• Conjugacy in Garside groups II: structure of the ultra summit set (2006, background): “3.1 Vertices and arrows in the USS graph of a rigid element . . . . . . . . . . . . . . 18 3.2 Paths in the USS graph of a rigid element . . . . . . . . . . . . . . . . . . . . .”
• Basic results on braid groups (2010, methodology): “From the practical point of view, there is a project explained in [10], to try to solve the conjugacy problem in B n in polynomial time, which starts by distinguishing the periodic, pseudo-Anosov and reducible (non-periodic) case.”