The Riemann-Hilbert mapping for mathfrak{sl}₂ -systems over genus two curves

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 $\mathrm{SL}_2(\mathbb{C})$.