Fibering Hyperbolic Manifolds and Hyperbolic Groups

In this thesis, we present new results regarding hyperbolic manifolds that fiber over the circle, and some consequences these have on hyperbolic groups. In particular, we construct a 5-dimensional hyperbolic manifold that fibers over the circle, and other hyperbolic manifolds ranging from dimension 4 to 8 that algebraically fiber. Our method, inspired by the work of Jankiewicz, Norin, and Wise, and relying on Bestvina-Brady theory, begins with a hyperbolic right-angled polytope. It involves a colouring of its facets, a state, and a set of moves, ultimately producing a hyperbolic manifold M with a map to the circle. This method is applied to a family of polytopes previously described by Potyagailo and Vinberg.Subsequently, we explore some consequences in terms of finiteness properties of the subgroups of hyperbolic groups. In particular, we exhibit a hyperbolic group that has a finite type subgroup that is not hyperbolic.These results are joint work with Bruno Martelli and Matteo Migliorini.