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.