Continuity of the bending map

Lecuire, Cyril

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 1, pp. 93-119.

L’application de plissage d’une variété hyperbolique de dimension $3$ associe à une métrique hyperbolique convexe cocompacte sur une variété compacte à bord sa lamination géodésique mesurée de plissage. Il a été démontré dans [KeS] et [KaT] que cette application est continue. Dans ce texte, on étudie l’extension de cette application à l’espace des métriques hyperboliques géométriquement finies. On introduit une relation d’équivalence dans l’espace des laminations géodésiques mesurées et on montre que l’application quotient de l’application de plissage est continue.

The bending map of a hyperbolic $3$-manifold maps a convex cocompact hyperbolic metric on a $3$-manifold with boundary to its bending measured geodesic lamination. As proved in [KeS] and [KaT], this map is continuous. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the space of measured geodesic laminations and show that the quotient map obtained from the bending map is continuous.

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