The subgroups of direct products of surface groups
Martin R. Bridson, James Howie, Charles F. Miller III,
Preprint, August 2000; now appeared in
Geom. Dedicata, vol. 92 (2002), 95--103.
A subgroup of a product of $n$ surface groups
is of type $FP_n$ if and only if it contains a subgroup
of finite index that is itself a product of (at most $n$)