By Margulis’ Lemma, a finite-volume complete hyperbolic n-manifold has a finite number of ends called cusps, each of which is diffeomorphic to the product of a flat (n-1)-manifold with the half-line. These flat manifolds are called cusp sections, and their possible configurations on hyperbolic manifolds is still very little understood. For instance, it was still not known if a hyperbolic manifold could have only rational homology spheres as cusp sections. In the 4-dimensional case, of the 10 flat 3-manifold diffeomorphism types, only the Hansche-Wendt manifold is a rational homology sphere, and it was conjectured if there existed an orientable hyperbolic 4-manifold such that all the cusps sections were such a manifold. We will introduce combinatorial tools to build manifolds by gluing copies of polytopes - a technique called colouring - and computational tools to tree-search for manifolds, thus providing an example that answers affirmatively this conjecture.

Joint work with Leone Slavich and Sasha Kolpakov.