2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141347If $G_1,...,G_n$ are limit groups and $S\subset G_1\times...\times G_n$ is of type $\FP_n(\mathbb Q)$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.20 pages, no figures. Final version. Accepted by the Annals of MathematicsGroup TheoryGeometric Topology20F65, 20F67, 20J05text