Quaternionic modular groups

Abstract

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 H n+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 H 4 or H 5 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 H 4 and H 5. 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.