  Hamiltonian structures for spatially compact locally homogeneous vacuum
universes are investigated, provided that the set of dynamical variables
contains the \Teich parameters, parameterizing the purely global geometry. One
of the key ingredients of our arguments is a suitable mathematical expression
for quotient manifolds, where the universal cover metric carries all the
degrees of freedom of geometrical variations, i.e., the covering group is
fixed. We discuss general problems concerned with the use of this expression in
the context of general relativity, and demonstrate the reduction of the
Hamiltonians for some examples. For our models, all the dynamical degrees of
freedom in Hamiltonian view are unambiguously interpretable as geometrical
deformations, in contrast to the conventional open models.
