From Mobius to Gyrogroups

Abstract

The evolution from Mobius to gyrogroups began in 1988, and is still ongoing in [14, 15]. Gyrogroups, a natural generalization of groups, lay a fruitful bridge between nonassociative algebra and hyperbolic geometry, just as groups lay a fruitful bridge between associative algebra and Euclidean geometry. More than 150 years have passed since the German mathematician August Ferdinand Mobius first studied the transformations that now bear his name. Yet, the rich structure he thereby exposed is still far from being exhausted, as the evolution from Mobius to gyrogroups demonstrates.