# Convex core

In geometry and topology , the **convex core** (English **convex core** , French **âme convex** ) plays an important role, especially in the theory of hyperbolic 3-manifolds.

## definition

Let it be a Riemannian manifold or a CAT (0) -space . The convex kernel is a minimal non-empty convex subset for which the inclusion is a homotopy equivalent .

## 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.

## literature

- 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