Pleating coordinates for the Earle embedding

Abstract

L'accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/ ∼ annales/), implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ (*) Recu le 27 octobre 1999, accepté le 28 fevrier 2001 (1)

Cusps in complex boundaries of one-dimensional Teichmüller space

“…Indeed, in the cases X = B G or E, this can be proved by combining the Local Pleating Theorem due to L. Keen and C. Series (cf. [26] see also Section 12) and the fact that all pleating rays are embedded arcs (see also Theorem 5.1 of [30] and Theorem 7.4 of [44]). Notice that the case of X = M had already been proven directly in Lemma 5.5 of [24].…”

“…Then, w(p/q) ∈ π 1 can be defined for all p/q ∈Q and the homology class of a simple closed curve in w(p/q) is equal to that in a −p b q . See for instance, [24], [30], [50], and [ where h runs over quasiconformal mappings h of R to R which are homotopic to f • f −1 and K(h) is the maximal dilation of h. This infimum is always attained by the unique extremal quasiconformal mapping (cf. [21] Quasifuchsian space has the product structure as follows: Henceforth, for any 3-manifold M with boundary, the following orientation convention is applied to ∂M : A frame f on ∂M is positive if (f, n) is positive with respect to the orientation of M where n is an inward-pointing vector on ∂M .…”

“…Details of the contents of this subsection can be found in [15], [24], [30], and [44]. Recall that for any point x ∈ X, a Kleinian group G x admits the distinguished invariant component Ω x , which is one of the three; Ω b ϕ , Ω e d , or Ω m µ .…”

Examples of pleating varieties for twice punctured tori

“…For this, ones needs the local pleating theorem ( [12], Theorem 8.1) and the limit pleating theorem ( [12], Theorem 5.1). The proofs of both these closely related results are much easier in the rational case (see [13], Theorem 3.7, and [16], Theorem 4.8), and extend without difficulty from once punctured tori to the general case. (The idea is to link each flat plane in ∂C to a disk in the regular set whose boundary contains the limit set of the Fuchsian subgroup which stabilises the plane in question.…”

Bers embedding of the Teichmüller space of a once-punctured torus

“…The reader should be careful with the term "slope" when referring to another paper devoted to the subject similar to ours because some authors prefer to call −q/p the slope of γ. The reason is explained by the fact that the pinching deformation of a once-punctured torus along the curve γ corresponds to letting the Teichmüller parameter τ ∈ H tend to the point −q/p ∈ ∂H (see [17] for details). In this article, however, we adopt r = p/q as the slope so that r represents the inclination of the vector (p, q).…”