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

Abstract

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 Étienne Ghys about Margulis' problem: existence of curves of negative Euler characteristic in compact quotients of ${\mathrm{SL}}_2(\mathbb{C})$.

35 pages; Accepted for publication at 'Bulletin de la Societé Mathématique de France'