Independent Loop Invariants for 2+1 Gravity

We identify an explicit set of complete and independent Wilson loop invariants for 2+1 gravity on a three-manifold $M=\mathbb{R}\times {\mathrm{\Sigma }}^g$, with ${\mathrm{\Sigma }}^g$ a compact oriented Riemann surface of arbitrary genus $g$. In the derivation we make use of a global cross section of the $PSU(1,1)$-principal bundle over Teichm"uller space given in terms of Fenchel-Nielsen coordinates.