Integers, Modular Groups, and Hyperbolic Space

In each of the normed division algebras over the real field \(\mathbb {R}\)—namely, \(\mathbb {R}\) itself, the complex numbers \(\mathbb {C}\), the quaternions IH, and the octonions \(\mathbb {O}\)—certain elements can be characterized as integers. An integer of norm 1 is a unit. In a basic system of integers the units span a 1-, 2-, 4-, or 8-dimensional lattice, the points of which are the vertices of a regular or uniform Euclidean honeycomb. A modular group is a group of linear fractional transformations whose coefficients are integers in some basic system. In the case of the octonions, which have a nonassociative multiplication, such transformations form a modular loop. Each real, complex, or quaternionic modular group can be identified with a subgroup of a Coxeter group operating in hyperbolic space of 2, 3, or 5 dimensions.