Cœur et nombre d'intersection pour les actions de groupes sur les arbres

We present the construction of a kind of convex core for the product of two actions of a group on R-trees. This geometric construction allows one to generalize and unify the intersection number of two curves or of two measured foliations on a surface, Scott's intersection number of two splittings, and the appearance of surfaces in Fujiwara-Papasoglu's construction of the JSJ splitting. In particular, this construction allows a topological interpretation of the intersection number analogous to the definition for curves in surfaces. As an application of this construction, we prove that an irreducible automorphism of the free group whose stable and unstable trees are geometric, is actually induced by a pseudo-Anosov homeomorphism of a surface.