Research on discrete groups and profective strucutures on Riemann surfaces

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A coordinate-system called lambda lengths is constructed for the SL(2,C) representation space of punctured surface groups. These lambda lengths can be considered as complexification of R.C.Penner's lambda lengths for decorated Teichmuller spaces. Via the coordinates the mapping class group acts on the representation space as a group of rational representations. We apply this fact to find hyperbolic 3-manifolds which fiber over the circle.