Continuity of the bending map

Abstract

The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. 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 shows that the quotient map obtained from the bending map is continuous. October 2025, an erratum has been added that corrects an error in the main Theorem. The topology used in the quotient space should not be the quotient topology but the "tubular topology" defined in this erratum.