Properness of the bending map
Cyril Lecuire
TL;DR
The paper analyzes the bending map from geometrically finite hyperbolic metrics on a compact 3-manifold $M$ to bending laminations on the boundary, introducing a tubular topology on $\mathcal{ML}(\partial M)/\mathcal{R}$ and studying the projection $b_{\mathcal{R}}$ of the bending map. It proves that $b_{\mathcal{R}}:\mathcal{GF}(M)\to\mathcal{P}(M)/\mathcal{R}$ is proper, establishing convergence of associated representations when bending laminations converge modulo $\mathcal{R}$ and upgrading this to convergence of boundary metrics and the limit geometry. The methods combine Thurston and Morgan–Shalen compactifications with cut-and-paste laminations, length estimates, and pleated-surface analysis to control both interior and boundary geometry. Additionally, the work shows that the mapping class group $Mod(M)$ acts properly discontinuously on the set of doubly incompressible laminations $\mathcal{D}(M)$, linking dynamic and geometric properties of the bending data to global deformation theory.
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.