Quaternionic modular groups Original Research Article

Matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n-space or hyperbolic (n+1)-space Hn+1. In particular, groups operating in the hyperbolic plane or hyperbolic 3-space may be represented by 2×2 matrices whose entries are rational integers or real or imaginary quadratic integers. The theory is extended here to groups operating in H4 or H5 and matrices over one of the three basic systems of quaternionic integers. Quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of H4 and H5. For four-dimensional groups the division ring of quaternions is treated as a Clifford algebra. Results in hyperbolic 5-space derive from the homeomorphism of inversive 4-space and the quaternionic projective line.