  We construct explicitly a (12g-12)-dimensional space P of unconstrained and
independent initial data for 't Hooft's polygon model of (2+1) gravity for
vacuum spacetimes with compact genus-g spacelike slices, for any g >= 2. Our
method relies on interpreting the boost parameters of the gluing data between
flat Minkowskian patches as the lengths of certain geodesic curves of an
associated smooth Riemann surface of the same genus. The appearance of an
initial big-bang or a final big-crunch singularity (but never both) is verified
for all configurations. Points in P correspond to spacetimes which admit a
one-polygon tessellation, and we conjecture that P is already the complete
physical phase space of the polygon model. Our results open the way for
numerical investigations of pure (2+1) gravity.
