Binary subgroups of direct products

We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties – the binary subgroups, B(Σ, µ) < G1 ×· · ·×Gm. These full subdirect products require strikingly few generators. If each Gi is finitely presented, B(Σ, µ) is finitely presented. When the Gi are non-abelian limit groups (e.g. free or surface groups), the B(Σ, µ) provide new examples of finitely presented, residually-free groups that do not have finite classifying spaces and are not of Stallings-Bieri type. These examples settle a question of Minasyan relating different notions of rank for residually-free groups. Using binary subgroups, we prove that if G1, . . . , Gm are perfect groups, each requiring at most r generators, then G1 × · · · × Gm requires at most rblog2 m + 1c generators