We prove in two different ways that the monodromy map from the space of irreducible $\mathfrak{ sl}_2$-differential-systems on genus two Riemann surfaces, towards the character variety of $\mathrm{ SL}_2$-representations of the fundamental group, is a local diffeomorphism. This is motivated by a question raised by \'Etienne Ghys about Margulis' problem: existence of curves of negative Euler characteristic in compact quotients of $\mathrm{ SL}_2(\mathbb{ C})$.
Categories: math.CV math.AG