Elementary embeddings in torsion-free hyperbolic groups

We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela's hyperbolic towers. Thus, if H embeds elementarily in a torsion free hyperbolic group Γ, we show that the group Γ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of H with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where Γ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.