  The classifying space of inertial reference frames in special relativity is
naturally hyperbolic. There is a remarkable interplay between central elements
of hyperbolic geometry and those of special relativity -- which, to a certain
extent, have already been observed in the past -- that we present and further
discuss in the paper. We aim at a geometrization of special relativity at the
level of kinematic space by giving to physical concepts/phenomena purely
geometric definitions/descriptions. In this way, the differences between
special relativity and classical mechanics can be seen as a manifestation of
the distinct geometric natures of their kinematic spaces.
