Abstract: We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions n<=5, a fact that leads us naturally to ask whether this may hold for any n. One consequence of this result is the existence of hyperbolic groups with finite-type subgroups that are not hyperbolic. The main tool is Bestvina - Brady theory applied to some hyperbolic n-manifolds that decompose very nicely into right-angled polytopes, enriched with the combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.