Martin R. Bridson, James Howie, Charles F. Miller III,
Preprint, April 2007. Accepted by
the Annals of Mathematics, October 2007.
A subgroup of a direct product of $n$ limit groups
is of type $FP_n(\Q)$ if and only if it contains a subgroup
of finite index that is itself a product of (at most $n$)
limit groups. This settles a conjecture of Sela.