Polyhedron complexes with simply transitive group actions and their realizations (Q1207399)

The aim of this article is to give a general method for dealing with the problems of discontinuous groups acting on \(d\)-dimensional spaces of constant curvature. The author is motivated by Poincaré's classical treatment of the analogous problems for the case when \(d < 3\), but his solutions are in the form of algorithms. He starts with a polyhedron \(\mathcal P\) with a finite flag structure. Then he proceeds to describe an algorithm for finding all possible identifications on the \((d - 1)\)-faces for which \(\mathcal P\) might be the fundamental domain for a group \(\mathcal G\) generated by the identifications. For each such group he constructs naturally a polyhedron \({\mathcal P}^{\mathcal G}\) on which \(\mathcal G\) acts simply transitively and such that \({\mathcal P}^{\mathcal G}/{\mathcal G}\) is \(\mathcal P\) modulo the identification. Finally he describes the basic algorithm along the lines of Poincaré's theorem for the case \(d = 2, 3\). It is the author's hope that this algorithmic approach would lend itself to computer use. He gives a few illustrations at the end.