  We consider gravity in 2+1 space-time dimensions, with negative cosmological
constant and a `Barbero-Immirzi' (B-I) like parameter, when the space-time
topology is of the form $ T^2 \times \mathbbm{R}$. The phase space structure,
both in covariant and canonical framework is analyzed. Full quantization of the
theory in the 'constrain first' approach reveals a finite dimensional physical
Hilbert space. An explicit construction of wave functions is presented. The
dimension of the Hilbert space is found to depend on the `Barbero-Immirzi' like
parameter in an interesting fashion. Comparative study of this parameter in
light of some of the recent findings in literarure for similar theories is
presented.
