Conjugacy in Garside groups I: cyclings, powers and rigidity

Abstract

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 ||∆|| 3 . In the particular case of braid groups {Bn, n ∈ 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.

The cyclic sliding operation in Garside groups

“…The corresponding result for SC(x) replaced by USS(x) was known to hold if the elements in USS(x) have canonical length greater than 1 [5], but there are counterexamples if the elements in USS(x) have canonical length equal to 1. The use of cyclic sliding allows us to drop the condition on the canonical length and hence yields a conceptually simpler result by removing the need to consider special cases.…”

“…For a detailed introduction to these groups, see [13]; a shorter introduction, containing all the details needed for this paper can be found in [5] ( §1.1 and the beginning of §1.2).…”

“…The methods for computing these sets, their properties, and the properties of their elements constitute the main topic of research concerning problems in Garside groups related to conjugacy. [15,8,17,19,5,6,7,22,23,24,26] In this paper we introduce a new operation, called cyclic sliding, and we propose to replace the usual cycling and decycling operations by this new one, as it is more natural from both the theoretical and computational points of view. Once cycling and decycling have been replaced by cyclic sliding, it is also natural to replace the ultra summit set USS(x) of an element x, whose definition is closely related to cycling, by its analogue for cyclic sliding.…”

“…In [5] the authors, together with Joan S. Birman, explained a project to solve the conjugacy problem in braid groups in polynomial time. This project, which was partially developed in [5,6,7], involved the concepts commonly used at that time, in particular ultra summit sets, cycling and decycling.…”

“…This project, which was partially developed in [5,6,7], involved the concepts commonly used at that time, in particular ultra summit sets, cycling and decycling. We remark that the results in this paper do not modify the essential ideas in the above project: one just replaces ultra summit sets by sets of sliding circuits, and cycling and decycling by cyclic sliding.…”