From Mobius to Gyrogroups

TL;DR

This paper discusses the development of gyrogroups from Mobius transformations, highlighting their role as a bridge between nonassociative algebra and hyperbolic geometry, extending Mobius's foundational work.

Contribution

It introduces gyrogroups as a natural generalization of groups, connecting nonassociative algebra with hyperbolic geometry, and traces their evolution from Mobius transformations.

Findings

Gyrogroups generalize groups and relate to hyperbolic geometry.

The evolution from Mobius transformations to gyrogroups demonstrates ongoing mathematical development.

Gyrogroups serve as a bridge between algebraic structures and geometric frameworks.

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.