# 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 . ${\ displaystyle M}$ ${\ displaystyle C (M)}$ ${\ displaystyle C (M) \ to M}$ ## 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 ${\ displaystyle {\ widetilde {M}}}$ ${\ displaystyle M = \ Gamma \ backslash {\ widetilde {M}}}$ ${\ displaystyle \ Lambda (\ Gamma) \ in \ partial _ {\ infty} {\ widetilde {M}}}$ ${\ displaystyle \ Gamma}$ ${\ displaystyle C (\ Lambda (\ Gamma)) \ subset {\ widetilde {M}}}$ ${\ displaystyle C (M) = \ Gamma \ backslash C (\ Lambda (\ Gamma))}$ .

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 ${\ displaystyle x \ in {\ widetilde {M}}}$ ${\ displaystyle r (x) \ in C (\ Lambda (\ Gamma))}$ ${\ displaystyle d (x, r (x)) = \ min \ left \ {d (x, y): y \ in C (\ Lambda (\ Gamma)) \ right \}}$ .

The mapping defined in this way can be continued into a continuous mapping . ${\ displaystyle r \ colon {\ widetilde {M}} \ to C (\ Lambda (\ Gamma))}$ ${\ displaystyle r \ colon {\ widetilde {M}} \ cup \ Omega (\ Gamma) \ to C (\ Lambda (\ Gamma))}$ ## 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. ${\ displaystyle \ delta}$ ${\ displaystyle \ delta> 0}$ ${\ displaystyle \ delta}$ The inclusion of the environment in is a deformation retract . ${\ displaystyle \ delta}$ ${\ displaystyle M}$ 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 . ${\ displaystyle \ delta}$ ${\ displaystyle \ Gamma \ backslash (H ^ {3} \ cup \ Omega (\ Gamma))}$ ${\ displaystyle \ Omega (\ Gamma)}$ ${\ displaystyle \ Gamma}$ ${\ displaystyle \ Gamma \ backslash \ Omega (\ Gamma)}$ ## 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. ${\ displaystyle \ Gamma}$ ${\ displaystyle {\ widetilde {M}}}$ ${\ displaystyle H ^ {n}}$ ${\ displaystyle M = \ Gamma \ backslash {\ widetilde {M}}}$ ${\ displaystyle \ delta}$ ## 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