Properness of the bending map

2025/10/08 by Lecuire, Cyril
#30F40 #57K32 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2510.07087

Abstract

The bending map of a hyperbolic 3-manifold with boundary maps a geometrically hyperbolic metric to its bending measured geodesic lamination. We show that the bending map is proper. As a byproduct of the proof we show that the group of isotopy classes of homeomorphisms of M acts properly discontinuously on the set of doubly incompressible measured geodesic laminations.

Citations

• Convex co-compact hyperbolic manifolds are determined by their pleating lamination
• Realisation of bending measured laminations by Kleinian surface groups
• Convergence of freely decomposable Kleinian groups
• Continuity of the bending map
• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Variations of the boundary geometry of 3--dimensional hyperbolic convex cores
• Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds

Related

• Continuity of the bending map
• Laminations in holomorphic dynamics
• Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds
• The induced metric and bending lamination on the boundary of convex hyperbolic 3-manifolds
• Variations of the boundary geometry of 3--dimensional hyperbolic convex cores