Revisiting Poincaré's Theorem on presentations of discontinuous groups via fundamental polyhedra

We give a new self-contained proof of Poincaré's Polyhedron Theorem on presentations of discon-
tinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based
on the theory of covering spaces, but only makes use of basic geometric concepts. In a sense one
hence obtains a proof that is of a more constructive nature than most known proofs.