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THE BEHAVIOUR OF FENCHELNIELSEN DISTANCE UNDER A CHANGE OF PANTS DECOMPOSITION
, 2011
"... Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition P and given a base complex structure X on S, there is an associated deformation space of complex structures on S, which we call the FenchelNielsen Teichmüller space associated to the pair ..."
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Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition P and given a base complex structure X on S, there is an associated deformation space of complex structures on S, which we call the FenchelNielsen Teichmüller space associated to the pair (P,X). This space carries a metric, which we call the FenchelNielsen metric, defined using FenchelNielsen coordinates. We studied this metric in the papers [1], [2] and [3], and we compared it to the classical Teichmüller metric (defined using quasiconformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding FenchelNielsen metrics is not necessarily biLipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.
On the inclusion of the quasiconformal Teichmüller space into the lengthspectrum Teichmüller space
, 2012
"... Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the ..."
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Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the lengthspectrum Teichmüller space. We work under this hypothesis that the basepoint is upperbounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the lengthspectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the lengthspectrum space. As a corollary we find an explicit description of the lengthspectrum Teichmüller space in terms of FenchelNielsen coordinates and we prove that the lengthspectrum Teichmüller space is pathconnected.
THURSTON’S METRIC ON TEICHMÜLLER SPACE AND ISOMORPHISMS BETWEEN FUCHSIAN GROUPS
, 2012
"... Abstract. The aim of this paper is to relate Thurston’s metric on Teichmüller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston’s asymmetric metric for surfaces with punctures. We also update some results ..."
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Abstract. The aim of this paper is to relate Thurston’s metric on Teichmüller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston’s asymmetric metric for surfaces with punctures. We also update some results of Sorvali on boundary isomorphisms of Fuchsian groups.