  In [1,2] we established and discussed the algebra of observables for 2+1
gravity at both the classical and quantum level. Here our treatment broadens
and extends previous results to any genus $g$ with a systematic discussion of
the centre of the algebra. The reduction of the number of independent
observables to $6g-6 (g > 1)$ is treated in detail with a precise
classification for $g = 1$ and $g = 2$.
