Emerging Topics working group | |

Topic: | One-relator groups, non-positive immersions and coherence |

Speaker: | Henry Wilton |

Affiliation: | Cambridge University |

Date: | Tuesday, March 26 |

Time/Room: | 4:00pm - 5:00pm/West Building Lecture Hall |

Video Link: | https://video.ias.edu/emergingtopics/2019/0326-HenryWilton |

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.