  We present the construction of some kind of "convex core" for the product of
two actions of a group on $\bbR$-trees. This geometric construction allows to
generalize and unify the intersection number of two curves or of two measured
foliations on a surface, Scott's intersection number of splittings, and the
apparition 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 a pseudo-Anosov homeomorphism on a surface.
