Subgroups of direct products of limit groups

Martin R. Bridson, James Howie, Charles F. Miller III, Hamish Short

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.