The recent series of results about one-relator groups (see, e.g. the previous two blog posts) culminates now in the final breakthrough: The conjecture of Baumslag from the 70s (that all one-relator groups are coherent) is resolved. The corresponding preprint by Marco Linton was put today on the arXiv:2303.05976.

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 … Continue reading "Homological coherence of one-relator groups"

Last month I blogged about coherent groups (i.e. groups whose finitely generated subgroups are finitely presented). There I also referred to an article of Daniel Wise about coherent groups that contains many open problems at the end. One of these problems is whether every one-relator group is coherent (a question posed by Baumslag in the … Continue reading "Virtually free-by-cyclic groups"

In this post we consider the Adams spectral sequence which computes the stable homotopy groups of spheres at the prime 2. The classes \(\{h_j\}_{j\ge 0}\) on the 1-line of the second page are called Hopf classes since they are related to the Hopf invariant: Adams proved that Hopf invariant one classes can only exist in … Continue reading "Differentials in the Adams spectral sequence"

Recall that a group G is called coherent if every finitely generated subgroup of G is finitely presented. Main examples are 3-manifold groups and (virtually) polycyclic groups. Instead of trying to motivate the study of coherent groups in my own words, let me instead just refer to Section 2 of the article `An Invitation to … Continue reading "Coherent groups"