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