Revisiting Poincar\'e's Theorem on presentations of discontinuous groups via fundamental polyhedra

TL;DR

This paper provides a new, constructive proof of Poincaré's Polyhedron Theorem for discontinuous groups of isometries on constant curvature Riemann manifolds, avoiding covering space theory.

Contribution

It introduces a self-contained, geometric proof that is more constructive than traditional methods, expanding understanding of group presentations via fundamental polyhedra.

Findings

New geometric proof of Poincaré's Theorem

Proof avoids covering space theory

Enhanced constructive approach to group presentations

Abstract

We give a new self-contained proof of Poincar\'e's Polyhedron Theorem on presentations of discontinuous 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.