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Binary Subgroups of Direct Products

We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\em binary subgroups}, $B(Σ,μ)<G_1\times\dots\times G_m$. These full subdirect products require strikingly few generators. If each $G_i$ is finitely presented, $B(Σ,μ)$ is finitely presented. When the $G_i$ 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 $G_1,\dots,G_m$ are perfect groups, each requiring at most $r$ generators, then $G_1\times\dots\times G_m$ requires at most $r \lfloor \log_2 m+1 \rfloor$ generators.