Convex hull of the Limes set
For manifolds with negative section curvature , the convex core can alternatively be defined as follows. It is the universal superposition , i.e. for a discrete group of isometries . Let be the Limes set of in the sphere at infinity and its convex hull . Then
The convex hull of the Limes set is therefore the universal superposition of the convex core.
For each point there is a unique point with
The mapping defined in this way can be continued into a continuous mapping .
Edge of the convex core
The edge of the convex kernel is generally not a smooth manifold. In the case of hyperbolic 3-manifolds, the edge of the convex core is as a folded sheet (engl .: pleated surface ), respectively. One therefore often considers a neighborhood of the convex core for a (arbitrary) . The edge of the neighborhood is a smooth manifold.
The inclusion of the environment in is a deformation retract .
In the case of hyperbolic 3-manifolds the -surrounding of the convex kernel is homeomorphic to the Klein manifold , where the discontinuity area denotes the action of on the sphere at infinity. In particular, in this case the edge of the convex core is homeomorphic to .
Convex-cocompact and geometrically finite groups
A discrete group of isometries of a CAT (0) -space (for example the hyperbolic space ) is called convex-cocompact , if the convex kernel is compact. It is called geometrically finite if one (i.e. every) neighborhood of the convex core has finite volume.
- William P. Thurston: The Geometry and Topology of Three-Manifolds online
- Matsuzaki, Katsuhiko; Taniguchi, Masahiko: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. ISBN 0-19-850062-9
- Canary, RD; Epstein, DBA; Green, PL: Notes on notes of Thurston. With a new foreword by Canary. London Math. Soc. Lecture Note Ser., 328, Cambridge Univ. Press, Cambridge, 2006.
- Epstein, DBA; Marden, A .: Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. Fundamentals of hyperbolic geometry: selected expositions, 117–266, London Math. Soc. Lecture Note Ser., 328, Cambridge Univ. Press, Cambridge, 2006.
- Bridgeman, Martin; Canary, Richard D .: The Thurston metric on hyperbolic domains and boundaries of convex hulls. Geom. Funct. Anal. 20 (2010), no. 6, 1317-1353. pdf