Subgroups of direct products of limit groups - Bridson, MR; Howie, J; Iii, C; Short, H (2007)
If $G_1,...,G_n$ are limit groups and $S\subset G_1×...× G_n$ 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.