The cycling operation endows the super summit set S x of any element x of a Garside group G with the structure of a directed graph Γ x . We establish that the subset U x of S x consisting of the circuits of Γ x can be used instead of S x for deciding conjugacy to x in G, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic approach to the conjugacy search problem in Garside groups. The results have implications for the security of recently proposed cryptosystems based on the hardness of problems related to the conjugacy (search) problem in braid groups.  2005 Elsevier Inc. All rights reserved.

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.

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.

The variation in genome arrangements among bacterial taxa is largely due to the process of inversion. Recent studies indicate that not all inversions are equally probable, suggesting, for instance, that shorter inversions are more frequent than longer, and those that move the terminus of replication are less probable than those that do not. Current methods for establishing the inversion distance between two bacterial genomes are unable to incorporate such information. In this paper we suggest a group-theoretic framework that in principle can take these constraints into account. In particular, we show that by lifting the problem from circular permutations to the affine symmetric group, the inversion distance can be found in polynomial time for a model in which inversions are restricted to acting on two regions. This requires the proof of new results in group theory, and suggests a vein of new combinatorial problems concerning permutation groups on which group theorists will be needed to collaborate with biologists. We apply the new method to inferring distances and phylogenies for published Yersinia pestis data.

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.