Hyperbolic four-manifolds

This is a short survey on finite-volume hyperbolic four-manifolds. We first describe some general theorems, and then focus on the geometry of the concrete examples that we found in the literature. The starting point of most constructions is an explicit reflection group Γ acting on H4, together with its Coxeter polytope P. Hyperbolic manifolds then arise either algebraically from the determination of torsion-free subgroups of Γ, or more geometrically by assembling copies of P. We end the survey by raising a few open questions