Convex-co-compact group

In the mathematical theory of Klein's groups , convex-cocompact groups are a generalization of cocompact lattices with numerous applications in the theory of dynamic systems , low-dimensional topology and harmonic analysis.

They can be considered not only in the theory of Klein's groups, but also more generally in the theory of discrete isometric groups of non-positively curved spaces.

definition

A discrete group of isometric drawings of a CAT (0) -space (for example the hyperbolic space ) is convex-kokompakt when the convex core of compact. ${\displaystyle \Gamma }$ ${\displaystyle {\widetilde {M}}}$ ${\displaystyle H^{n}}$ ${\displaystyle M=\Gamma \backslash {\widetilde {M}}}$

Convex-cocompact Klein groups

For Klein groups , the following conditions are equivalent: ${\displaystyle \Gamma \subset Isom^{+}(H^{3})=PSL(2,\mathbb {C} )}$

• ${\displaystyle \Gamma }$ is convex-cocompact.
• Every point of the Limes set is a conical boundary point .
• Each point of the Limes set is a horospheric boundary point .
• The Klein manifold is compact. ${\displaystyle (H^{3}\cup \Omega (\Gamma ))/\Gamma }$

Convex-co-compact groups in higher rank

Let be a symmetric space of non-compact type . ${\displaystyle {\widetilde {M}}=G/K}$

If is irreducible and of rank , then the cocompact lattices are the only convex-cocompact subsets of . ${\displaystyle {\widetilde {M}}}$ ${\displaystyle \geq 2}$ ${\displaystyle G}$

For any symmetric spaces of non-compact type , for every convex-co-compact subgroup there are decompositions and with for , so that for each there is either or a co-compact lattice in . ${\displaystyle {\widetilde {M}}=G/K}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G=G_{1}\times \ldots \times G_{r}}$ ${\displaystyle \Gamma =\Gamma _{1}\times \ldots \times \Gamma _{r}}$ ${\displaystyle \Gamma _{i}\subset G_{i}}$ ${\displaystyle i=1,\ldots ,r}$ ${\displaystyle i}$ ${\displaystyle rank(G_{i}/K_{i})=1}$ ${\displaystyle \Gamma _{i}}$ ${\displaystyle G_{i}}$

Therefore, in the higher-ranked symmetric space theory, RCA groups are studied as a richer generalization of convex-cocompact groups.

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.

Individual evidence

1. ↑ B. Kleiner, B. Leeb: Rigidity of invariant convex sets in symmetric spaces. Invent. Math. 163 (2006), no. 3, 657-676