Combinatorial Model Manifolds for Kleinian Surface Groups from Curve Complex Hierarchies
A way to reconstruct the large-scale geometry of an infinite-volume hyperbolic 3-manifold from purely combinatorial data. Starting from a bi-infinite geodesic in the complex of curves of a surface and the hierarchy of subsurface geodesics thickening it, one glues standard blocks and solid-torus tubes into a model manifold whose short curves and Margulis tubes match those of the actual manifold via a uniformly Lipschitz map. Students learn how curve-complex combinatorics, quasiconvexity of bounded-length curves, and subsurface projection bounds combine to give a priori length bounds and a combinatorial criterion for bounded geometry.
Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds (Warwick lectures)
Edited and revised lecture notes from a workshop on Kleinian Groups and Hyperbolic 3-Manifolds held in Warwick in 2001. The paper covers both combinatorial and geometrical aspects of hyperbolic 3-man…