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

Gabriel Calsamiglia; Bertrand Deroin; Viktoria Heu; Frank Loray

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

Complex Variables 2018-12-03 v4 Algebraic Geometry

Authors: Gabriel Calsamiglia , Bertrand Deroin , Viktoria Heu , Frank Loray

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

Keywords

diffeomorphism groups holomorphic mappings algebraic topology

Cite

@article{arxiv.1602.02273,
  title  = {The Riemann-Hilbert mapping for $\mathfrak{sl}_2$ -systems over genus two curves},
  author = {Gabriel Calsamiglia and Bertrand Deroin and Viktoria Heu and Frank Loray},
  journal= {arXiv preprint arXiv:1602.02273},
  year   = {2018}
}

Comments

35 pages; Accepted for publication at 'Bulletin de la Societ\'e Math\'ematique de France'

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

Abstract

Keywords

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