The sectional curvature is one of the main invariants of Riemannian manifolds. But despite its importance, there is actually little known about questions like: What is the distinction (for closed manifolds) between admitting a metric of positive sectional curvature vs admitting a metric of nonnegative sectional curvature? Another problem is also to distinguish positive sectional … Continue reading "Positive vs nonnegative sectional curvature"

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"