1 Sapienza University of Rome, Via dei Marsi, 78- 00185 Rome, Italy
The properties of the group PGL(2,C) on the Upper Poincar´e Half Plane have been analyzed.
In particular, the classification of points and geodesics has been achieved by considering the solution to the free Hamiltonian associated problem.
The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincar´e Half Plane is found.
A definition of quadratic surds is proposed, for which the folding group corresponds to the tiling group, (also) for Hamiltonian systems on the Hyperbolic Plane (also realized as the Upper Poincar´e Half Plane (UPHP)).
The initial conditions determine the result of the folding of the trajectories as tiling punctured tori and for tori.
Keywords: 02.20.-a Group theory mathematics, 43-02 Abstract harmonic analysis 02.20.Rt Discrete subgroups of Lie groups, 00A69 General applied mathematics, 70S05 Lagrangian formalism, Hamiltonian formalism, Upper Poincar´e Half Plane (UPHP).
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