  The most general solution to the Einstein equations in $4=3+1$ dimensions in
the asymptotical limit close to the cosmological singularity under the BKL
(Belinski-Khalatnikov-Lifshitz) hypothesis, for which space gradients are
neglected and time derivatives only are considered, can be visualized by the
behavior of a billiard ball in a triangular domain on the Upper Poincar\'e Half
Plane (UPHP). The behavior of the billiard system (named 'big billiard') can be
schematized by dividing the succesions of trajectories according to Poincar\'e
return map on the sides of the billiard table, according to the paradigms
implemented by the BKL investigation and by the CB-LKSKS (Chernoff- Barrow-
Lifshitz- Khalatnikov- Sinai- Khanin- Shchur) one. Different maps are obtained,
according to different symmetry-quotienting mechanisms used to analyze the
dynamics according to the symmetries of the billiard domain and to the features
of the geodesics on the UPHP. In the inhomogenous case, new structures have
been uncovered, such that, in this framework, the billiard table (named 'small
billiard') consists of 1/6 of the previous one. The connections between the
symmetry-quotienting mechanisms are further investigated on the UPHP. The
relation between the complete billiard and the small billiard are also further
explained according to the role of Weyl reflections. The quantum properties of
the system are sketched as well, and the physical interpretation of the
wavefunction is further developped. In particular, a physical interpretation
for the symmetry-quotienting maps is proposed, according to the Fourier
decomposition of the energy levels of the wavefunction of the universe, as far
as the meaning of epochs and eras is concerned from a quantum point of view.
