Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession : The Theory of Gyrogroups and Gyrovector Spaces Thomas Precession: The Missing Link Gyrogroups: Modeled on Einstein’S Addition The Einstein Gyrovector Space Hyperbolic Geometry of Gyrovector Spaces The Ungar Gyrovector Space The MÖbius Gyrovector Space Gyrogeometry Gyrooprations — the SL(2 c) Approach The Cocycle Form The Lorentz Group and its Abstraction The Lorentz Transformation Link Other Lorentz Groups. Evidence that Einsteins addition is regulated by the Thomas precession has come to light turning the notorious Thomas precession previously considered the ugly duckling of special relativity theory into the beautiful swan of gyrogroup and gyrovector space theory where it has been extended by abstraction into an automorphism generator called the Thomas gyration. The Thomas gyration in turn allows the introduction of vectors into hyperbolic geometry where they are called gyrovectors in such a way that Einsteins velocity additions turns out to be a gyrovector addition. Einsteins addition thus becomes a gyrocommutative gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular the gyrovector space with gyrovector addition given by Einsteins (Möbius) addition forms the setting for the Beltrami (Poincaré) ball model of hyperbolic geometry. The gyrogroup theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einsteins 1905 view of the Lorentz length contraction which was contradicted in 1959 by Penrose Terrell and others. The application of gyrogroup theoretic techniques clearly tilt the balance in favor of Einstein.