The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Gallo, Daniel M.; Kapovich, Michael; Marden, Albert

doi:10.2307/121044

Cited by 137 publications

(233 citation statements)

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“…Mess cites Gallo's announcement [52] for this, though the complete proof was not made available until several years later, in joint work with Marden and M. Kapovich [54]. Gallo's proposed proof of Theorem N.6.2 is based on the existence of a pants decomposition of S with the property that the holonomy of each pair of pants is quasi-Fuchsian.…”

Section: N6 the Case Of De Sitter Spacementioning

confidence: 99%

Notes on a paper of Mess

et al. 2007

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Section: N6 the Case Of De Sitter Spacementioning

confidence: 99%

Notes on a paper of Mess

et al. 2007

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“…is proper [GKM,§11.4] [Tan2] and injective [Kra]. Within V (S) there is the closed set AH(S) of discrete and faithful representations, and its interior QF (S), which consists of quasi-Fuchsian representations.…”

Section: Applications: Holonomy and Fuchsian è 1 Structuresmentioning

confidence: 99%

The Schwarzian derivative and measured laminations on Riemann surfaces

Dumas

2007

Duke Math. J.

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Abstract. A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation, which can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be considered as the Schwarzian derivative of a È 1 structure, to which one can naturally associate another measured geodesic lamination using grafting.We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomorphism ML (S) → Q(X) for each conformal structure X on a compact surface S. We show that these maps are nearly the same, differing by a multiplicative factor of −2 and an error term of lower order than the maps themselves (which we bound explicitly).As an application we show that the Schwarzian derivative of a È 1 structure with Fuchsian holonomy is close to a 2π-integral JenkinsStrebel differential. We also study compactifications of the space of È 1 structures using the Schwarzian derivative and grafting coordinates; we show that the natural map between these extends to the boundary of each fiber over Teichmüller space, and we describe this extension.

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“…230, §50], with a modern history developed by many authors ( [Ma69], [He75], [Fa83], [ST83], [Go87], [GKM00], [Ta97], [Mc98]). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the "Grafting Conjecture".…”

Section: §1 Introductionmentioning

confidence: 99%

The grafting map of Teichmüller space

Scannell

Wolf

2002

J. Amer. Math. Soc.

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IntroductionOne of the underlying principles in the study of Kleinian groups is that aspects of the complex projective geometry of quotients ofĈ by these groups reflect properties of the three-dimensional hyperbolic geometry of the quotients of H 3 by these groups. Yet, even though it has been over thirty-five years since Lipman Bers exhibited a holomorphic embedding of the Teichmüller space of Riemann surfaces in terms of the projective geometry of a Teichmüller space of quasi-Fuchsian manifolds, no corresponding embedding in terms of the three-dimensional hyperbolic geometry has been presented. One of the goals of this paper is to give such an embedding. This embedding is straightforward and has been expected for some time ([Ta97], [Mc98]): to each member of a Bers slice of the space QF of quasi-Fuchsian 3-manifolds, we associate the bending measured lamination of the convex hull facing the fixed "conformal" end.The geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting, an operation on projective structures on surfaces that traces its roots back at least to Klein [Kl33, p. 230 ). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the "Grafting Conjecture". This conjecture states that for a fixed measured lamination λ, the self-map of Teichmüller space induced by grafting a surface along λ is a homeomorphism of Teichmüller space; our contribution to this argument is a proof of the injectivity of the grafting map. While the principal application of this result that we give is to geometric coordinates on the Bers slice of QF , one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of Kleinian groups (see §5.2); thus we feel that this result is of interest in its own right.We now state our results and methods more precisely. Throughout, S will denote a fixed differentiable surface which is closed, orientable, and of genus g ≥ 2. Let T g be the corresponding Teichmüller space of marked conformal structures on S, and let P g denote the deformation space of (complex) projective structures on S (see §2 for definitions).

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The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Cited by 137 publications

References 34 publications

Notes on a paper of Mess

Notes on a paper of Mess

The Schwarzian derivative and measured laminations on Riemann surfaces

The grafting map of Teichmüller space

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