2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76401The proof of the Tits alternative for $Out(F_n)$ is completed. The main tool is a Kolchin type theorem, proved in this paper. It states that a finitely generated subgroup of $Out(F_n)$ consisting of unipotent automorphisms can be conjugated into an upper-triangular subgroup (this is interpreted via train-tracks).50 pagesGeometric Topologytext