Abstract :
The group of all holomorphic automorphisms of the complex unit disk consists of
M¨obius transformations involving translation-like holomorphic automorphisms and
rotations. The former are called gyrotranslations. As opposed to translations of the
complex plane, which are associative-commutative operations i.e., their composi-
tion law is associative and commutative. forming a group, gyrotranslations of the
complex unit disk fail to form a group. Rather, left gyrotranslations are gyroasso-
ciative-gyrocommutative operations i.e., their composition law is gyroassociative
and gyrocommutative. forming a gyrogroup. The complex unit disk gyrogroup has
previously been studied by the author Aequationes Math. 47, 1994, 240]254..
Employing analogies shared by complex numbers and linear transformations of
vector spaces, we extend in this article the complex disk gyrogroup and its M¨obius
transformations into the ball of any real inner product space and its generalized
M¨obius transformations. A gyrogroup is a mathematical object which first arose in
the study of relativistic velocities which, under velocity addition, form a nongroup
gyrogroup, as opposed to prerelativistic velocities, which form a group under
velocity addition. It has been discovered that the mathematical regularity, seemingly
lost in the transition from prerelativistic to relativistic velocities, is concealed
in a relativistic effect known as Thomas precession. In its abstract context, Thomas
precession is called Thomas gyration, giving rise to our ‘‘gyroterminology.’’ Our
gyroterminology, developed by the author Amer. J. Phys. 59, 1991, 824]834.,
involves terms like gyrogroups, gyroassociative-gyrocommutative laws, and gyroautomorphisms,
in which we extensively use the prefix ‘‘gyro.’’
