An Algorithm for Classification of Fundamental Polygons for a Plane Discontinuous Group

Abstract

Based on the procedure given in [15] we describe an algorithm, implemented in a computer program, for complete enumeration of combinatorial equivalence classes of fundamental polygons for any fixed plane discontinuous group given by its signature. This is a solution of a long standing problem, we call it Poincaré-Delone problem to honour of Henry Poincaré and Boris Nikolaevich Delone (Delaunay). We give examples and computations together with some complete lists of combinatorially different polygons which serve as fundamental domains for the groups with the Macbeath signatures, e.g. (2,+,[];{}), (3,-,[];{}) and (3,+,[];{}).