Characters of sl(2) representations of groups

Given a compact orientable surface Σ, let S (Σ) be the set of isotopy classes of essential unoriented simple loops in the surface. We determine a complete set of relations for a function defined on S(Σ) to a field K to be the character of an SL(2, K) representations. Furthermore, the relations are supported in the 1-holed torus and the 4-holed sphere subsurfaces. This establishes that Grothendieck’s reconstruction principle is valid for SL(2, K)-character varieties of surface groups. As a consequence, we obtain an explicit description of the set of all characters of SL(2, K) representations.