On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type

Alessandrini, Daniele; Liu, Lixin; Papadopoulos, Athanase; Su, Weixu; Sun, Zhongguo

doi:10.48550/arxiv.1003.0980

Cited by 11 publications

(23 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We shall consider Fenchel-Nielsen coordinates for spaces of homotopy classes of hyperbolic structures on S. We carried out in [2] a study of these parameters in the setting of surfaces of infinite type. These parameters are associated to a fixed base hyperbolic structure equipped with a fixed geodesic pair of pants decomposition P = {C i } i∈I .…”

Section: The Fenchel-nielsen Teichmüller Spacesmentioning

confidence: 99%

“…In this paper, we investigate various Teichmüller spaces associated to a surface of infinite topological type, in continuation of works that were done in [10] and [2]. An initial impulse to these works was given by a paper by H. Shiga [15].…”

Section: Introductionmentioning

confidence: 99%

“…bi-Lipschitz homeomorphism constants, or it can be equal to a sup norm associated to Fenchel-Nielsen coordinates, and so on. We refer to the papers [10] and [2] for a discussion of such ideas. The Teichmüller spaces we obtain have consequently different names, and in this paper, we shall deal with the so-called "length-spectrum Teichmüller space", "quasiconformal Teichmüller space" and "Fenchel-Nielsen Teichmüller space".…”

Section: Introductionmentioning

confidence: 99%

“…Such a phenomenon can be seen in examples of Basmajian, in his paper [3]. In the paper [2], we gave a necessary and sufficient condition (which we called Nielsen-convexity) under which given a hyperbolic structure on a surface of infinite type, a topological pair of pants decomposition (or, equivalently, any topological pair of pants decomposition) can be made geodesic. Consequently, when we shall talk about Fenchel-Nielsen coordinates for a hyperbolic surface, we shall tacitly assume that the underlying hyperbolic structure is Nielsen-convex.…”

Section: Introductionmentioning

confidence: 99%

“…In the paper [2], we also introduced the notion of a Fenchel-Nielsen Teichmüller space, T F N (H 0 ), based at a hyperbolic surface H 0 , with its associated Fenchel-Nielsen metric d F N , relative to a fixed geodesic pants decomposition P of H 0 .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On various Teichmüller spaces of a surface of infinite topological type

Alessandrini

Liu

Papadopoulos

et al. 2011

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: The Fenchel-nielsen Teichmüller Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On various Teichmüller spaces of a surface of infinite topological type

Alessandrini

Liu

Papadopoulos

et al. 2011

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

On local comparison between various metrics on Teichmüller spaces

et al. 2011

View full text Add to dashboard Cite

International audienceThere are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint ( a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller space, and there are others. In general, these spaces are set-theoretically different. An important question is therefore to understand relations between these spaces. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between these spaces. In this paper, we obtain local metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of finite type. We recall that in the case of a surface of finite type, all these Teichmüller spaces coincide setwise. In the case of a surface of finite type with no boundary components (and possibly with punctures), we show that the restriction of the identity map to any thick part of Teichmüller space is globally bi-Lipschitz with respect to the length spectrum metric and the classical Teichmüller metric on the domain and on the range respectively. In the case of a surface of finite type with punctures and boundary components, there is a metric on the Teichmüller space which we call the arc metric, whose definition is analogous to the length spectrum metric, but which uses lengths of geodesic arcs instead of lengths of closed geodesics. We show that the restriction of the identity map restricted to any ``relative thick" part of Teichmüller space is globally bi-Lipschitz, with respect to any of the three metrics: the length spectrum metric, the Teichmüller metric and the arc metric on the domain and on the range

show abstract