Structure and Finiteness Properties of Subdirect Products of Groups

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 direct summands. 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. A useful structure theory for subgroups of finite direct products of groups has yet to be developed. To begin to study such subgroups it is natural to assume one knows about the subgroups of the direct factors and to concentrate on subdirect products. Recall that G is termed a subdirect product of the groups A1, . . . , An if G ⊆ A1 × · · · × An is a subgroup that projects surjectively to each factor. Work by various authors has exposed the surprisingly rich structure to be found amongst the subdirect products of superficially-tame groups. For example, in contrast to the fact that subdirect products of abelian or nilpotent groups are again in the specified class, nonabelian free groups harbour a great diversity of subdirect products, including some with unsolvable decision problems [19]. This diversity has long been known, but it is only as a result of more recent work by Baumslag-Roseblade [5] and Bridson-Howie-Miller-Short [9] that it has been understood as a phenomenon that is intimately tied to the failure of various homological finiteness conditions. Date: 11 April 2007. 1991 Mathematics Subject Classification. 20F05, 20E07, 20F67, 20J05, 20F10.