Injectivity Radius of Representations of Triangle Groups and Planar Width of Regular Hypermaps

Abstract

 We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing upper bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type with a given planar width.