Binary subgroups of direct products (Q6113474)

Summary: We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the \textit{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 \(G_1, \ldots, G_m\) 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.