Ohshika, Ken’ichi
Reduced Bers boundaries of Teichmüller spaces  [ Les bords réduits de Bers des espaces de Teichmüller ]
Annales de l'institut Fourier, Tome 64 (2014) no. 1 , p. 145-176
MR 3330544 | Zbl 06387269
doi : 10.5802/aif.2842
URL stable : http://www.numdam.org/item?id=AIF_2014__64_1_145_0

Classification:  30F40,  30F60,  57M50
Mots clés: bord de Bers, espace de Teichmüller, groupe kleinien
Nous considérons un espace quotient du bord de Bers de l’espace de Teichmüller, lequel nous appelons le bord réduit de Bers, en identifiant les points dans chaque espace des déformations quasi-conformes sur le bord de Bers. Nous démontrons que ce bord est indépendant du point de base, et que l’action du groupe modulaire s’y étend continûment. Ce théorème est une réponse affirmative à une conjecture de Thurston. Il a aussi conjecturé que ce bord est homéomorphe à l’espace des laminations non-mesurées. Cette conjecture-ci a besoin de correction : la topologie quotient du bord réduit de Bers est différente de la topologie induite de l’espace des laminations non-mesurées. En plus, nous démontrons que tout auto-homéomorphisme du bord réduit de Bers est induit par une unique classe d’applications de la surface. Nous aussi donnons un moyen de déterminer la limite dans le bord pour une suite donnée dans l’espace de Teichmüller.
We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space lying there into a point.This boundary turns out to be independent of the basepoint, and the action of the mapping class group extends continuously to this boundary.This is an affirmative answer to Thurston’s conjecture.He also conjectured that this boundary is homeomorphic to the unmeasured lamination space by the correspondence coming from ending laminations.This part of the conjecture needs some correction: we show that the quotient topology of the reduced Bers boundary is different form the topology induced from the unmeasured lamination space.Furthermore, we show that every auto-homeomorphism on the reduced Bers boundary comes from a unique extended mapping class.We also give a way to determine the limit in the reduced Bers boundary for a given sequence in Teichmüller space.

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