  We give the description of discretized moduli spaces (d.m.s.) $\Mcdisc$
introduced in \cite{Ch1} in terms of a discrete de Rham cohomologies for each
moduli space $\Mgn$ of a genus $g$, $n$ being the number of punctures. We
demonstrate that intersection indices (cohomological classes) calculated for
d.m.s. coincide with the ones for the continuum moduli space $\Mc$ compactified
by Deligne and Mumford procedure. To show it we use a matrix model technique.
The Kontsevich matrix model is a generating function for these indices in the
continuum case, and the matrix model with the potential $N\alpha \tr {\bigl(-
\fr 12 \L X\L X +\log (1-X)+X\bigr)}$ is the one for d.m.s. In the last case
the effects of reductions become relevant, but we use the stratification
procedure in order to express integrals over open spaces $\Mdisc$ in terms of
intersection indices which are to be calculated on compactified spaces. The
coincidence of the cohomological classes for both continuum and d.m.s. models
enables us to propose the existence of a quantum group structure on d.m.s. Then
d.m.s. are nothing but cyclic (exceptional) representations of a quantum group
related to a moduli space $\Mc$. Considering the explicit expressions for
integrals of Chern classes over $\Mc$ and $\Mcdisc$ we conjecture that each
moduli space $\Mc$ in the Kontsevich parametrization can be presented as a
coset $\Mc ={\bf T}^d/G$, $d=3g-3+n$, where ${\bf T}^d$ is some
$d$--dimensional complex torus and $G$ is a finite order symmetry group of
${\bf T}^d$.
