The Riemann-Hilbert mapping for ${\mathfrak{s}\mathfrak{l}}_2$ -systems over genus two curves

We prove in two different ways that the monodromy map from the space of irreducible ${\mathfrak{s}\mathfrak{l}}_2$-differential-systems on genus two Riemann surfaces, towards the character variety of ${\mathrm{S}\mathrm{L}}_2$-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 ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$.