Subdirect products of groups and the n-(n + 1)-(n + 2) Conjecture

We analyse the subgroup structure of direct products of groups. Ear-lier work on this topic has revealed that higher finiteness properties play a crucial role in determining which groups appear as subgroups of direct products of free groups or limit groups. Here, we seek to relate the finite-ness properties of a subgroup to the way it is embedded in the ambient product. To this end we formulate a conjecture on finiteness properties of fibre products of groups. We present different approaches to this con-jecture, proving a general result on finite generation of homology groups of fibre products and, for certain special cases, results on the stronger finiteness properties Fn and FPn.