One-relator groups, non-positive immersions and coherence

Abstract: There seems to be an analogy between the classes of fundamental groups of compact 3-manifolds and of one-relator groups.  (Indeed, many 3-manifold groups are also one-relator groups.) For instance, Dehn’s Lemma for 3-manifolds (proved by Papakyriakopoulos) can be seen as analogous to Magnus’ Freiheitssatz for one-relator groups.  But the analogy is still very incomplete, and since there are deep results on each side that have no analogue on the other, there is a strong incentive to flesh it out.

Coherence is one property for which the analogy remains unknown.  A group is *coherent* if every finitely generated subgroup is finitely presented.  A famous theorem of Scott asserts that 3-manifold groups are coherent; Baumslag asked in 1974 if one-relator groups are coherent, and the question remains open.

In this talk, I’ll describe some recent progress towards Baumslag’s problem, which centres around Wise’s notion of *non-positive immersions*.   We will see that one-relator groups are homologically coherent, that one-relator groups with torsion are coherent, and that low-rank subgroups of one-relator groups are always free.