  We explore an elementary construction that produces finitely presented groups
with diverse homological finiteness properties -- the {\em binary subgroups},
$B(\Sigma,\mu)<G_1\times\dots\times G_m$. These full subdirect products require
strikingly few generators. If each $G_i$ is finitely presented, $B(\Sigma,\mu)$
is finitely presented. When the $G_i$ are non-abelian limit groups (e.g. free
or surface groups), the $B(\Sigma,\mu)$ 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.

Group Theory