Binary subgroups of direct products

Abstract

We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties – the binary subgroups, $B(\Sigma ,\mu )<G_1\times \cdots \times G_m$. 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 $G_1\times \cdots \times G_m$ requires at most $r\lfloor {\log }_{2}m+1\rfloor$ generators.