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arXiv · 1602.02273

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

Gabriel Calsamiglia·Bertrand Deroin·Viktoria Heu·Frank Loray
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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 $\mathrm{SL}_2(\mathbb{C})$.

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Gabriel Calsamiglia, Bertrand Deroin, Viktoria Heu, Frank Loray. 2016-02-06. The Riemann-Hilbert mapping for $\mathfrak{sl}_2$ -systems over genus two curves. https://arxiv.org/abs/1602.02273

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