The aim of this paper is to show that the fundamental group of an oriented 3manifold which satisfies Thurston's geometrization conjecture, has a solvable conjugacy problem. In other terms, for any such 3-manifold M , there exists an algorithm which can decide for any couple of elements u, v of π 1 (M ) whether u and v are in the same conjugacy class of π 1 (M ) or not. More topologically, the algorithm decides for any two loops in M , whether they are freely homotopic or not.

Abstract. We establish sufficient conditions for the C * -simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their non-trivial subnormal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions.Much of our analysis deals with conditions on an action of a group Γ on a tree T which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree T and on its boundary ∂T , and faithfulness in a strong sense. An important step in this analysis is to identify automorphism of T which are slender, namely such that their fixed-point sets in ∂T are nowhere dense for the shadow topology.

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