We investigate the structure of subdirect products
of groups, particularly their finiteness properties. We pay special attention
to the subdirect products of free groups, surface groups
and HNN extensions. We prove that a finitely presented subdirect product
of free and surface groups virtually contains a term of the lower central
series of the direct product or else fails to intersect one of the
This leads to a characterization of the finitely
presented subgroups of the direct product of 3 free or surface groups,
and to a solution to the conjugacy problem for arbitrary finitely
presented subgroups of direct products of surface groups.
We obtain a
formula for the first homology of a subdirect product of two free groups
and use it to show there is no algorithm to determine the first homology
of a finitely generated subgroup.