Binary Subgroups of Direct Products

TL;DR

This paper introduces binary subgroups of direct products, providing new finitely presented examples with diverse homological properties, and establishes bounds on the number of generators needed for certain product groups.

Contribution

It constructs binary subgroups with novel properties, including examples that are finitely presented yet lack finite classifying spaces, and proves bounds on generators for products of perfect groups.

Findings

Binary subgroups can be finitely presented with few generators.

New examples of residually-free groups without finite classifying spaces.

Bound on generators for products of perfect groups based on their number and properties.

Abstract

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.