Abstract: Let S be a compact two dimensional surface with boundary. We define two moduli spaces closely related to representations of the fundamental group of S to a split algebraic group G, e.g. G=SL(m). We equip them with distinguished collections of coordinate systems such that the corresponding transition functions are subtraction free. Thus we can define the points of the corresponding moduli spaces with coefficients in positive numbers, or, more generally, in any semifield.

For G=SL(2) we recover this way the classical Teichmuller spaces and, taking points with values in the tropical semifields, Thurston's laminations on S. For general G we get a generalisation of the Teichmuller-Thurston theory. We quantize these spaces using the quantum dilogarithm.

This is joint work with V. Fock, available as math.AG/0311149.