Monodromy groups of CP1‐structures on punctured surfaces

Gupta, Subhojoy

doi:10.1112/topo.12189

Cited by 7 publications

(4 citation statements)

References 31 publications

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“…When is a representation 𝜌 ∶ 𝜋 1 𝑆 → Aff(ℂ) the holonomy of a branched affine structure, when the number and order of branch points are prescribed? It turns out that for any such branched structure, the Schwarzian derivative of the developing map has a pole of order 2 at each branch point, and thus can also be thought of as a meromorphic projective structure, whose holonomy representations have been recently studied in [8,13].…”

Section: Additional Remarks and Further Questionsmentioning

confidence: 99%

Translation surfaces and periods of meromorphic differentials

Chenakkod

Gianluca

Gupta

2022

Proceedings of London Math Soc

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Let S$S$ be an oriented surface of genus g$g$ and n$n$ punctures. The periods of any meromorphic differential on S$S$, with respect to a choice of complex structure, determine a representation χ:Γg,n→double-struckC$\chi :\Gamma _{g,n} \rightarrow \mathbb {C}$ where normalΓg,n$\Gamma _{g,n}$ is the first homology group of S$S$. We characterise the representations that thus arise, that is, lie in the image of the period map sans-serifPer:normalΩMg,n→sans-serifHom(normalΓg,n,C)$\textsf {Per}:\Omega \mathcal {M}_{g,n}\rightarrow \textsf {Hom}(\Gamma _{g,n}, {\mathbb {C}})$. This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S$S$ with the prescribed holonomy χ$\chi$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

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Section: Additional Remarks and Further Questionsmentioning

confidence: 99%

Translation surfaces and periods of meromorphic differentials

Chenakkod

Gianluca

Gupta

2022

Proceedings of London Math Soc

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“…Poincaré himself asked it in the case where Σ is a punctured sphere, see [15,Paragraph 4]. Very recently, Gupta announced an answer for every punctured surface [10]. In [7], Gallo, Kapovich and Marden provided a complete answer for closed surfaces.…”

Section: Introductionmentioning

confidence: 99%

Pentagon representations and complex projective structures on closed surfaces

Fils¹

2023

Annales De l'Institut Fourier

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We define a class of representations of the fundamental group of a closed surface of genus 2 to PSL 2 (C): the pentagon representations. We show that they are exactly the non-elementary PSL 2 (C)-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to PSL 2 (C) is the holonomy of a projective structure, possibly with one branched point of order 2. We show that pentagon representations arise as such holonomies and repair their proof.Résumé. -Nous définissons une classe de représentations du groupe fondamental d'une surface fermée de genre 2 dans PSL 2 (C) : les représentations pentagones. Nous montrons que ce sont exactement les représentations d'un groupe de surface dans PSL 2 (C) qui n'admettent pas de décomposition de Schottky, i.e. de décomposition en pantalons telle que la restriction de la représentation à chaque pantalon est un isomorphisme sur un groupe de Schottky. Ce faisant, nous exhibons une lacune dans la preuve de Gallo, Kapovich et Marden du fait que toute représentation non-élémentaire d'un groupe de surface dans PSL 2 (C) est l'holonomie d'une structure projective, avec éventuellement un point de branchement d'ordre 2. Nous montrons que les représentations pentagones sont de telles holonomies et réparons leur preuve.

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“…When is a representation ρ : π 1 S → Aff(C) the holonomy of a branched affine structure, when the number and order of branch-points are prescribed? It turns out that for any such branched structure the Schwarzian derivative of the developing map has a pole of order two at each branch-point, and thus can also be thought of as a meromorphic projective structure, whose holonomy representations have been recently studied in [Gup19], [FG]. This paper also does not address the problem of understanding a "holonomy fiber" beyond whether it is empty or not.…”

Section: Introductionmentioning

confidence: 99%

Translation surfaces and periods of meromorphic differentials

Chenakkod,

Faraco,

Gupta

2021

Preprint

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Let S be an oriented surface of genus g and n punctures. The periods of any meromorphic differential on S, with respect to a choice of complex structure, determine a representation χ : Γg,n → C where Γg,n is the first homology group of S. We characterize the representations that thus arise, that is, lie in the image of the period map Per : ΩMg,n → Hom(Γg,n, C). This generalizes a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S with the prescribed holonomy χ. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data. Contents 1. Introduction 1 2. Geometry of translation surfaces 7 3. Surgeries on translation surfaces 9 Part I. Translation structures with prescribed holonomy 11 4. Some preliminaries 12 5. How to geometrize handles 12 6. How to geometrize punctured spheres 15 7. Geometrizing open surfaces 17 Part II. Translation structures in a prescribed stratum 22 8. Trivial holonomy I: Punctured Spheres 22 9. Trivial holonomy II: Positive genus surfaces 28 10. Spheres with non-trivial holonomy 35 11. Positive genus surfaces with non-trivial holonomy 38 12. The case of rational holonomy 44 Appendix A. Periods of meromorphic differentials with prescribed poles 54 Appendix B. Proof of Lemma 10.3 55 References 57

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Monodromy groups of CP1‐structures on punctured surfaces

Cited by 7 publications

References 31 publications

Translation surfaces and periods of meromorphic differentials

Translation surfaces and periods of meromorphic differentials

Pentagon representations and complex projective structures on closed surfaces

Translation surfaces and periods of meromorphic differentials

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