Recently appeared a highly interesting paper by Italiano, Martelli, Migliorini on the arXiv (2105.14795) (Okay, this was already three weeks ago – I don’t follow the “geometric topology” – and someone had to point me there). The paper solves at least two longstanding open problems by studying an explicit 5-dimensional hyperbolic manifold.

It is well-known that there are hyperbolic 3-manifolds which fibre over the circle. The first examples were given by Jørgensen in 1977. In fact, from the seminal work of Agol and Wise we know by now that every hyperbolic 3-manifold has a finite sheeted covering which fibres over the circle. Interestingly, it seems there was not a single example of a hyperbolic manfifold of dimension greater than 3 which fibres over the cirlce. In their paper Italiano, Martelli, Migliorini show that there is a 40-cusped hyperbolic 5-manifold which indeed fibres.

Filling the cusps, they can also solve a famous open problem in geometric group theory. They construct a hyperbolic group \(G\) and a subgroup \(H \leq G \) of finiteness type (F) such that \(H\) itself is not a hyperbolic group. In fact \(H\) is the fundamental group of a 4-dimensional aspherical finite complex.

Koji Fujiwara already posted a follow-up paper on the arXiv (2106.08549) which (builing on their example) exhibits a closed 5-manifold of nonpositive curvature which fibres. The fundamental group is hyperbolic relative to subgroups isomorphic to \(\mathbb{Z}^3\) and the fibre is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to any abelian subgroups.