Quadratic Integers and Coxeter Groups
Norman W. Johnson, Asia Ivić Weiss · Canadian Journal of Mathematics · 1999
Abstract Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversiven-space or hyperbolic (n+1)-space Hn+1. For smalln, thesemay be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of Hn+1. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group PSL2( ), the Gaussian modular (Picard) group PSL2( [i]), and the Eisenstein modular group PSL2( [ω]).