**Authors:** A. A. Ungar

### Abstract

In this era of an increased interest in loop theory, the Einstein velocity addition law has fresh resonance. One of the most fascinating aspects of recent work in Einstein's special theory of relativity is the emergence of special grouplike loops. The special grouplike loops, known as gyrocommutative gyrogroups, have thrust the Einstein velocity addition law, which previously has operated mostly in the shadows, into the spotlight. We will find that Einstein addition is a gyrocommutative gyrogroup operation that forms the setting for the Beltrami-Klein (Poincare) ball model of hyperbolic geometry just as the common vector addition is a commutative group operation that forms the setting for the standard model of Euclidean geometry. The resulting analogies to which the grouplike loops give rise lead us to new results in (i) hyperbolic geometry; (ii) relativistic physics; and (iii) quantum information and computation.

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