Conjugacy in Garside groups III: Periodic braids

Abstract

An element in Artin's braid group B n is said to be periodic if some power of it lies in the center of B n . In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms.This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B n , which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.

The cyclic sliding operation in Garside groups

“…Although SC(x) is equal to RSSS(x) in most cases, the case ℓ s (x) = 1 in which the sets differ is not irrelevant. For instance, in the braid group B n , a periodic braid x which is not conjugate to a power of ∆ has summit length 1, but the conjugacy problem for such braids is far from being an easy issue [7].…”

“…In [7], the authors and Joan S. Birman showed that the number of elements in the ultra summit sets of some periodic braids in B n is exponential in n. More precisely, |USS(δ)| = 2 n−2 , where δ = σ n−1 · · · σ 1 ∈ B n . To overcome this difficulty, in [7] we also gave a polynomial algorithm to solve the conjugacy search problem for all periodic braids (which of course does not involve computing the whole ultra summit set).…”

“…Since SC(x) is contained in USS(x) for every x ∈ B n , and it is in general smaller for elements of canonical length one (like δ above), one may wonder whether SC(δ) has polynomial size, allowing to use the general algorithm given in this paper (and in [20]) instead of the particular one given in [7], which only works for periodic braids. Unfortunately the answer is negative: There are only two elements which are in USS(δ) but not in SC(δ), as it is shown in the following result.…”

“…In [7,Proposition 10] one can find a characterisation of the elements of USS(δ): They are those simple braids s whose associated permutation is a single cycle of length n of the form…”

“…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.…”