The isometry group of $n$-dimensional Einstein gyrogroup

Abstract

The space of $n$-dimensional relativistic velocities normalized to $c=1$, $$\mathbb{B}=\{\mathbf{v}\in {\mathbb{R}}^n:\Vert \mathbf{v}\Vert <1\},$$ is naturally associated with Einstein velocity addition ${\oplus }_E$, which induces the rapidity metric $d_E$ on $\mathbb{B}$ given by $d_E(\mathbf{u},\mathbf{v})={\tanh }^{-1}\Vert -\mathbf{u}{\oplus }_E\mathbf{v}\Vert$. This metric is also known as the Cayley-Klein metric. We give a complete description of the isometry group of $(\mathbb{B},d_E)$, along with its composition law.