Hyperbolic 4-manifolds, perfect circle-valued Morse functions and infinitesimal rigidity

In the first part, we exhibit some (compact and cusped) finite-volume hyperbolic 4-manifolds M with perfect circle-valued Morse functions, that is Morse functions from M to the circle with only index 2 critical points. The main tools used are the colourings and the states on right-angled polytopes. We construct in particular one example where every generic circle-valued function is homotopic to a perfect one. A consequence is the existence of infinitely many finite-volume (compact and cusped) hyperbolic 4-manifolds M having a handle decomposition with bounded numbers of 1- and 3-handles, so with bounded Betti numbers b1(M), b3(M) and rank of the fundamental group. In the second part we prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. The 4-dimensional examples are the ones described in the first part. The 5-dimensional example is diffeomorphic to N×R for some aspherical 4-manifold N which does not admit any hyperbolic structure, and was discovered by Italiano-Martelli-Migliorini. To this purpose we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.