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.

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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.

Let A and A be two Artin groups of spherical type, and let A 1 , . . . , A p (resp. A 1 , . . . , A q ) be the irreducible components of A (resp. A ). We show that A and A are commensurable if and only if p = q and, up to permutation of the indices, A i and A i are commensurable for every i. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed n, we give a complete classification of the irreducible Artin groups of rank n that are commensurable with the group of type A n . Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability. Proposition 2. Let Γ and Ω be two Coxeter graphs of spherical type. If A[Γ] and A[Ω] are commensurable, then Γ and Ω have the same number of vertices. Proof. Suppose that A[Γ] and A[Ω] are commensurable. Let n be the number of vertices of Γ and let m be the number of vertices of Ω. We know that the cohomological dimension of A[Γ] is n and the cohomological dimension of A[Ω] is m (Paris, 2004, Proposition 3.1). As every finite index subgroup of A[Γ] has the same cohomological dimension as A[Γ] and every finite index subgroup of A[Ω] has the same cohomological dimension as A[Ω], we have n = m.

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.