Applications of global coordinate systems for the representation spaces of punctured surface groups

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R.C.Penner’s lambda length coordinate system for the Teichmuller space of a punctured surface has many applications to complex analysis, differential geometry, topology and to mathematical physics. We introduced a complexification of Penner’s coordinates in order to parametrize SL(2,C) representation spaces of punctured surface groups. The complex coordinates are applied to the theories of Kleinian groups, hyperbolic 3-manifolds, mapping class groups, complex dynamics and number theory.