The riemann-hilbert mapping for sl2 systems over genus two curves

Gabriel Calsamiglia; Bertrand Deroin; Viktoria Heu; Frank Loray

doi:10.24033/bsmf2778

The riemann-hilbert mapping for sl2 systems over genus two curves

Gabriel Calsamiglia, Bertrand Deroin, Viktoria Heu, Frank Loray

Research output: Contribution to journal › Article › peer-review

Abstract

We prove in two different ways that the monodromy map from the space of irreducible sl2 differential systems on genus two Riemann surfaces, towards the character variety of SL2 representations of the fundamental group, is a local diffeomorphism. We also show that this is no longer true in the higher genus case. Our work is motivated by a question raised by Étienne Ghys about Margulis' problem: The existence of curves of negative Euler characteristic in compact quotients of SL2(C).

Original language	English
Pages (from-to)	159-195
Number of pages	37
Journal	Bulletin de la Societe Mathematique de France
Volume	147
Issue number	1
DOIs	https://doi.org/10.24033/bsmf2778
Publication status	Published - 1 Jan 2019
Externally published	Yes

Keywords

Foliations
Holomorphic connections
Monodromy
Projective structures
Riemann-Hilbert
sl2 systems over curves

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10.24033/bsmf2778

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title = "The riemann-hilbert mapping for sl2 systems over genus two curves",

abstract = "We prove in two different ways that the monodromy map from the space of irreducible sl2 differential systems on genus two Riemann surfaces, towards the character variety of SL2 representations of the fundamental group, is a local diffeomorphism. We also show that this is no longer true in the higher genus case. Our work is motivated by a question raised by {\'E}tienne Ghys about Margulis' problem: The existence of curves of negative Euler characteristic in compact quotients of SL2(C).",

keywords = "Foliations, Holomorphic connections, Monodromy, Projective structures, Riemann-Hilbert, sl2 systems over curves",

author = "Gabriel Calsamiglia and Bertrand Deroin and Viktoria Heu and Frank Loray",

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AU - Deroin, Bertrand

AU - Heu, Viktoria

AU - Loray, Frank

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Y1 - 2019/1/1

N2 - We prove in two different ways that the monodromy map from the space of irreducible sl2 differential systems on genus two Riemann surfaces, towards the character variety of SL2 representations of the fundamental group, is a local diffeomorphism. We also show that this is no longer true in the higher genus case. Our work is motivated by a question raised by Étienne Ghys about Margulis' problem: The existence of curves of negative Euler characteristic in compact quotients of SL2(C).

AB - We prove in two different ways that the monodromy map from the space of irreducible sl2 differential systems on genus two Riemann surfaces, towards the character variety of SL2 representations of the fundamental group, is a local diffeomorphism. We also show that this is no longer true in the higher genus case. Our work is motivated by a question raised by Étienne Ghys about Margulis' problem: The existence of curves of negative Euler characteristic in compact quotients of SL2(C).

KW - Foliations

KW - Holomorphic connections

KW - Monodromy

KW - Projective structures

KW - Riemann-Hilbert

KW - sl2 systems over curves

U2 - 10.24033/bsmf2778

DO - 10.24033/bsmf2778

M3 - Article

AN - SCOPUS:85065616381

SN - 0037-9484

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EP - 195

JO - Bulletin de la Societe Mathematique de France

JF - Bulletin de la Societe Mathematique de France

IS - 1

ER -

The riemann-hilbert mapping for sl2 systems over genus two curves

Abstract

Keywords

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