  It is considered a model of compact Riemann surface in genus two, represented
geometrically by two-parametric hyperbolic octagon with an order four
automorphism and described algebraically by the corresponding Fuchsian group.
Introducing the Fenchel--Nielsen variables, we compute the Weil--Petersson (WP)
symplectic two-form for parameter space and analyze the closed isoperimetric
orbits of octagons. WP-Area in parameter space and the canonical action--angle
variables for the orbits are found. Exploiting the ideas from the loop quantum
gravity, we generate relativistic kinematics by the Lorentz boost and quantize
WP-area. We treat the evolution in terms of global variables within the "big
bounce" concept.
