Locally indicable groups outside Brodskiĭ's quasivariety

Abstract

We provide a set of group properties that together obstruct membership in the quasivariety that Brodskiĭ constructed in his proof that torsion-free one-relator groups are locally indicable. There are examples of locally indicable groups with these properties, which implies that Brodskiĭ's quasivariety is strictly contained in the quasivariety of all locally indicable groups. This disproves a conjecture of Howie.