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

TL;DR

This paper demonstrates that the monodromy map for irreducible rak{sl}_2-differential systems on genus two Riemann surfaces is a local diffeomorphism, advancing understanding of the geometric structures related to rak{sl}_2$-systems.

Contribution

The authors prove the local diffeomorphism property of the monodromy map for genus two curves using two different methods, addressing a question related to Margulis' problem.

Findings

Monodromy map is a local diffeomorphism for genus two surfaces.

Provides two different proofs of the diffeomorphism property.

Connects the result to questions about negative Euler characteristic 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 \'Etienne Ghys about Margulis' problem: existence of curves of negative Euler characteristic in compact quotients of $\mathrm{SL}_2(\mathbb{C})$.