| Summary: | 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.
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