Building hyperbolic manifolds using Coxeter polytopes

In this work we explicitly build original examples of cusped hyperbolic manifolds gluing Coxeter polytopes. In detail, we realize every closed flat 3-manifold as a cusp section of a complete, finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps. Moreover, we show that for every such 3-manifold, a dense subset of its flat metrics can be realized as cusp sections of a cusp-transitive 4-manifold. Furthermore, we prove that there are a lot of 4-manifolds with pairwise isometric cusps, for any given cusp type. Finally, we build a non-compact, orientable, hyperbolic 4-manifold of finite volume that does not admit any spin structure.