On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space

Daniele Alessandrini (1) , Lixin Liu (2) , Athanase Papadopoulos (3) , Weixu Su (3, 4)

1 Département de Mathématiques, Université de Fribourg
2 Department of Mathematics
3 IRMA - Institut de Recherche Mathématique Avancée
4 Fudan University [Shanghai]

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 length-spectrum Teichmüller space. We work under this hypothesis that the basepoint is upper-bounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the length-spectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the length-spectrum space. As a corollary we find an explicit description of the length-spectrum Teichmüller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmüller space is path-connected.

Mots clés: Fenchel-Nielsen coordinates.; quasiconformal map; Teichmüller space; Fenchel-Nielsen coordinates; Length-spectrum metric; quasiconformal metric

Domaines: Topologie géométrique [math.GT]

Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su. On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space. Monatshefte für Mathematik, 2016, 179 (2), pp.165-189. ⟨10.1007/s00605-015-0813-9⟩. ⟨hal-00664093⟩