  In [1,2] we established and discussed the algebra of observables for $2+1$
gravity at both the classical and quantum level, and gave a systematic
discussion of the reduction of the expected number of independent observables
to $6g - 6 (g > 1)$. In this paper the algebra of observables for the case
$g=2$ is reduced to a very simple form. A Hilbert space of state vectors is
defined and its representations are discussed using a deformation of the
Euler-Gamma function. The deformation parameter $\th$ depends on the
cosmological and Planck's constants.
