The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx 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 appr...

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphi...

It is proved that for any prime p a finitely generated nilpotent group is conjugacy separable in the class of finite p-groups if and only if the tor-sion subgroup of it is a finite p-group and the quotient group by the torsion subgroup is abelian. 1. Let K be a class of groups. A group G is called residual K (or K-residual) if for each non-unit element a ∈ G there is a homomorphism ϕ of G onto ...

Small Gaussian groups are a natural generalization of spherical Artin groups in which the existence of least common multiples is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be of Coxeter type. We show here how to extend the Elrifai-Morton solution for the conjugacy problem in braid groups to every small Gaussian group.

We describe a significantly improved algorithm for computing the conjugacy classes of a finite permutation group with trivial soluble radical. We rely on existing methods for groups that are almost simple, and we are concerned here only with the reduction of the general case to the almost simple case. Dedicated to Charles Leedham-Green on the occasion of his 65th birthday

There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x−1Ax = B for some x ∈ Γ with ‖x‖ less than a linear function of max{‖γ‖ : γ ∈ A∪B}. (The coefficients of this linear function depend only on k and δ.) These results have i...

In this paper we study the conjugacy problem in polycyclic groups. Our main result is that we create polycyclic groups Gn whose conjugacy problem is at least as hard as the subset sum problem with n indeterminates. As such, the conjugacy problem over the groups Gn is NP-complete where the parameters of the problem are taken in terms of n and the length of the elements given on input.