Homological coherence of one-relator groups

Seems to be an interesting time right now to be working on coherent groups (previous blog posts: link and link). Recall that coherent groups are those whose finitely generated subgroups are finitely presented, and that coherence of one-relator groups is one of the main open problems.

It is known that one-relator groups with torsion elements are coherent, so the remaining case is the one of torsion-free one-relator groups. Just recently, there was progress made on this by Andrei Jaikin (link to preprint): He proved that torsion-free one-relator groups are homologically coherent!

Homological coherence is a weaker property than coherence: It means that every finitely generated subgroup is of type \(\text{FP}_2\) (a definition and discussion of this property may be found on Wikipedia: link). Note that it was for a long time an open problem whether every group of type \(\text{FP}_2\) is actually finitely presentable. The first counter-example to this (i.e. a group of type \(\text{FP}_2\) but not finitely presentable) was given by Bestvina and Brady (link) in 1997. So being homologically coherent is very close to being coherent.