On the inclusion of the quasiconformal Teichmüller space into the lengthspectrum Teichmüller space

Alessandrini, Daniele
Département de Mathématiques, Université de Fribourg, Switzerland

Liu, L.
Department of Mathematics, Sun YatSen University, Quangzhou, China

Papadopoulos, A.
Institut de Recherche Mathématique Avancée (Université de Strasbourg and CNRS), France

Su, W.
Institut de Recherche Mathématique Avancée (Université de Strasbourg and CNRS), France  Department of Mathematics, Fudan University, Shanghai, China
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Published in:
 Monatshefte für Mathematik.  2016, vol. 179, no. 2, p. 165–189
English
This paper is about surfaces of infinite topological type. Unlike the case of surfaces of finite type, there are several deformation spaces associated with a surface S of infinite topological type. Such spaces depend on the choice of a basepoint (that is, the choice of a fixed conformal structure or hyperbolic structure on S) and they also depend on the choice of a distance on the set of equivalence classes of marked hyperbolic structures. We address the question of the comparison between two deformation spaces, namely, the quasiconformal Teichmüller space and the lengthspectrum Teichmüller space. There is a natural inclusion map of the quasiconformal space into the lengthspectrum space, which is not always surjective. We work under the hypothesis that the basepoint (a hyperbolic surface) satisfies a condition we call “upperboundedness”. This means that this surface admits a pants decomposition defined by curves whose lengths are bounded above. The theory under this upper boundedness hypothesis shows a dichotomy. On the one hand there are surfaces satisfying what we call Shiga’s condition, i.e. they admit a pants decomposition defined by curves whose lengths are bounded above and below. If the base point satisfies Shiga’s condition, then the inclusion of the quasiconformal space into the length spectrum space is surjective, and it is a homeomorphism. In this paper we concentrate on the other kind of upperbounded surfaces, which we call “upperbounded with short interior curves”. This means that the corresponding hyperbolic surface admits a pants decomposition defined by curves whose lengths are bounded above, and such that the lengths of some interior curves approach zero. We show that in this case the behavior is completely different. Under this hypothesis, the image of the inclusion between the two Teichmüller spaces is nowhere dense in the lengthspectrum space. As a corollary of the methods used, we obtain an explicit parametrization of the lengthspectrum Teichmüller space in terms of Fenchel–Nielsen coordinates and we prove that the lengthspectrum Teichmüller space is pathconnected.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/304954
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