Holonomy map fibers of ℂ<i>P</i><sup>1</sup>-structures in moduli space

Baba, Shinpei; Gupta, Subhojoy

doi:10.1112/jtopol/jtv013

Cited by 7 publications

(22 citation statements)

References 17 publications

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“…The class up to conjugation of the holonomy representation therefore defines a geometric invariant of the structure. This invariant is known to be far from classifying; describing the set of surfaces having the same holonomy as been investigated in several contexts (see for instance [1] or [4] in the case of branched projective structures).…”

Section: Holonomymentioning

confidence: 99%

“…Recall that a branched affine structure on of holonomy ρ is a section of M ρ which is transverse to F ρ except at a finite number of points where it is tangent to the foliation at a finite order. 1 Consider an arbitrary trivialisation V × M ρ 0 of E above a neighbourhood V of ρ 0 . The graph of a section s 0 of M ρ 0 can be pushed to each M ρ for ρ ∈ V by means of this trivialisation.…”

Section: Proposition 33 the Subset Of Geometric Representations Is Amentioning

confidence: 99%

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Mapping class group dynamics and the holonomy of branched affine structures

Ghazouani

2017

Math. Z.

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Section: Holonomymentioning

confidence: 99%

Section: Proposition 33 the Subset Of Geometric Representations Is Amentioning

confidence: 99%

Mapping class group dynamics and the holonomy of branched affine structures

Ghazouani

2017

Math. Z.

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“…For marked projective structures on a closed surface, any holonomy fiber, if non-empty, is necessarily discrete, and has been studied in [Bab17] and [BG15]. In the case of translation structures on a closed surface, these fibers could have positive dimension, and define the isoperiodic foliations mentioned earlier in this Introduction.…”

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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“…For closed surfaces, the grafting description for projective structures has been useful in the study of the monodromy (or holonomy) map (5) hol : P g → χ g from P g to the PSL 2 (C)-character variety of surface-group representations (see, for example, [Bab17] and [BG15]).…”

Section: Introductionmentioning

confidence: 99%

Meromorphic projective structures, grafting and the monodromy map

Gupta¹,

Mj²

2019

Preprint

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A meromorphic projective structure on a punctured Riemann surface X \ P is determined, after fixing a standard projective structure on X, by a meromorphic quadratic differential with poles of order three or more at each puncture in P. In this article we prove the analogue of Thurston's grafting theorem for such meromorphic projective structures, that involves grafting crowned hyperbolic surfaces. This also provides a grafting description for projective structures on C that have polynomial Schwarzian derivatives. As an application of our main result, we prove the analogue of a result of Hejhal, namely, we show that the monodromy map to the decorated character variety (in the sense of Fock-Goncharov) is a local homeomorphism.

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Holonomy map fibers of ℂP¹-structures in moduli space

Cited by 7 publications

References 17 publications

Mapping class group dynamics and the holonomy of branched affine structures

Mapping class group dynamics and the holonomy of branched affine structures

Translation surfaces and periods of meromorphic differentials

Meromorphic projective structures, grafting and the monodromy map

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Holonomy map fibers of ℂP1-structures in moduli space

Cited by 7 publications

References 17 publications

Mapping class group dynamics and the holonomy of branched affine structures

Mapping class group dynamics and the holonomy of branched affine structures

Translation surfaces and periods of meromorphic differentials

Meromorphic projective structures, grafting and the monodromy map

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Holonomy map fibers of ℂP¹-structures in moduli space