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"

Let us consider the following classical problem from geometry (the case \(n=3\) is basically Plateau’s problem): Let \(\Gamma\) be a smooth, closed, oriented, \((n−1\))-dimensional submanifold of \(\mathbb{R}^{n+1}\). If we consider all the smooth, compact, oriented hypersurfaces \(M \subset \mathbb{R}^{n+1}\) with \(\partial M = \Gamma\), does there exist one with least area among them? In the … Continue reading "Regularity of minimizing hypersurfaces"

Recall the Soul Theorem of Cheeger and Gromoll: If \((M,g)\) is a complete and connected Riemannian manifold of non-negative sectional curvature, then there exists a closed, totally convex and totally geodesic embedded submanifold whose normal bundle is diffeomorphic to \(M\). Such a submanifold is called a soul of \((M,g)\) and the Riemannian metric induced on … Continue reading "Double Soul Conjecture"

The union-closed sets conjecture is the following extremely easy to state conjecture about subsets of finite sets: Assume that \(\mathcal{F}\) is a family of subsets of \(\{1, 2, \ldots, n\}\) which is union-closed; this means that for any two sets \(A,B\) in \(\mathcal{F}\) their union \(A \cup B\) is also a member of \(\mathcal{F}\). Then … Continue reading "Progress on the union-closed sets conjecture"