Quaternionic modular groups

✍ Scribed by Norman W. Johnson; Asia Ivić Weiss

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 250 KB
• Volume: 295
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00107-x

✦ Synopsis

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 r n1 . In particular, groups operating in the hyperbolic plane or hyperbolic 3-space may be represented by 2 Â 2 matrices whose entries are rational integers or real or imaginary quadratic integers. The theory is extended here to groups operating in r 4 or r 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 r 4 and r 5 . For four-dimensional groups the division ring of quaternions is treated as a Cliord algebra. Results in hyperbolic 5-space derive from the homeomorphism of inversive 4-space and the quaternionic projective line.