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\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.