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

TL;DR

This paper investigates the subgroup structure of direct products of groups, proposing a conjecture relating finiteness properties of fibre products to their embeddings, supported by new results on homology and finiteness properties.

Contribution

It formulates a new conjecture on finiteness properties of fibre products and provides partial proofs and approaches, advancing understanding of subgroup structures in direct products.

Findings

Proves a general result on finite generation of homology groups of fibre products.

Establishes results on finiteness properties F_n and FP_n for specific cases.

Proposes a conjecture linking subgroup embeddings to finiteness properties.

Abstract

We analyse the subgroup structure of direct products of groups. Earlier 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 finiteness 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 conjecture, proving a general result on finite generation of homology groups of fibre products and, for certain special cases, results on the stronger finiteness properties F_n and FP_n.