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

in Expositiones Mathematica (2015), 33(4), 401--430

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 ... [more ▼]

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. [less ▲]