Geometrically finite group

In geometry , the term geometrically finite group was originally used in 2- and 3-dimensional hyperbolic geometry as a designation for discrete groups of isometries that have a convex polyhedron with a finite number of sides as a fundamental domain . In the higher-dimensional hyperbolic geometry, more general definitions are used, which in the case of isometric groups of 2 or 3-dimensional space are equivalent to the original definition, but are more general in higher dimensions.

Every finitely generated discrete group of isometries of the hyperbolic plane is geometrically finite. In higher dimensions, lattices and convex-co-compact groups are examples of geometrically finite groups.

Isometric groups of 3-dimensional hyperbolic space (Klein groups)

A Klein group is called geometrically finite if it fulfills one of the following equivalent conditions.

• For each , the neighborhood of the convex kernel has finite volume. ${\displaystyle \delta >0}$ ${\displaystyle \delta }$
• For one , the neighborhood of the convex kernel has finite volume. ${\displaystyle \delta >0}$ ${\displaystyle \delta }$
• The thick part of the convex core is compact .
• For sufficiently small ones , the complement of the -cuspidal part in the convex nucleus is compact. ${\displaystyle \epsilon }$ ${\displaystyle \epsilon }$
• Each point of the limit set is a conical boundary point or a restricted parabolic fixed point .
• Every point of the limit set is a horospheric boundary point or a restricted parabolic fixed point.
• Every Dirichlet polyhedron is finite.
• There is a finite Dirichlet polyhedron.
• The Klein manifold is the union of a compact subspace with a finite set of standard vertices . ${\displaystyle (H^{3}\cup \Omega (\Gamma ))/\Gamma }$

Geometrically finite hyperbolic metrics on a given 3-manifold are uniquely determined by their conformal boundaries (i.e. the quotients of the regions of discontinuity in the sphere at infinity ).

Isometric groups of higher-dimensional hyperbolic spaces and of Hadamard manifolds

More generally, a discrete group of isometries of a Hadamard manifold is called geometrically finite if it satisfies one of the following equivalent conditions. ${\displaystyle X}$

• ${\displaystyle (X\cup \Omega (\Gamma ))/\Gamma }$ is the union of a compact subspace with a finite set of standard peaks.
• Each point of the limit set is a conical limit point or a bounded parabolic fixed point.
• The thick part of the convex core is compact.
• There is an upper bound for the order of finite subgroups and for one the -surrounding of the convex kernel has finite volume. ${\displaystyle \delta >0}$ ${\displaystyle \delta }$

For every finitely generated discrete group of isometries of the hyperbolic plane is geometrically finite and has a finite fundamental polyhedron, i.e. H. a (not necessarily compact) fundamental domain that is a polyhedron with finitely many sides. ${\displaystyle n=2}$

For a geometrically finite group does not necessarily have to have a finite fundamental polyhedron. For example there are geometrically finite groups with an infinite number of points . ${\displaystyle n\geq 4}$ ${\displaystyle \Gamma \subset Isom(H^{n})}$

Hyperbolic groups and convergence groups

For a convergence group acting on a compact, metric space , one defines geometric finiteness as follows: Every point from is a conical Limes point or a restricted parabolic fixed point. The terms “conical limit point” and “restricted parabolic fixed point” are intrinsically defined. A conical limit point is a point to which there is a sequence of different elements and points with and converges uniformly on compacts against the mapping, which is constant . A bounded parabolic fixed point is a point whose stabilizer is parabolic (i.e., infinite, leaves a point solid, and contains no loxodromic elements) and for which the quotient is compact. ${\displaystyle X}$ ${\displaystyle \Gamma }$ ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle \gamma _{n}\in \Gamma }$ ${\displaystyle a_{\pm }\in X}$ ${\displaystyle \lim _{n\to \infty }\gamma _{n}x=a_{+}}$ ${\displaystyle \gamma _{n}\mid _{X\setminus \left\{x\right\}}}$ ${\displaystyle a_{-}}$ ${\displaystyle x\in X}$ ${\displaystyle X}$ ${\displaystyle (X\setminus \left\{x\right\})/\Gamma _{x}}$

This definition can be applied in particular to hyperbolic groups , because these act as convergence groups on their edge at infinity.

Examples of geometrically finite Klein groups

• Quasi-fox groups
• Schottky groups

Conformal edge

The Isomorphismussatz of Marden reduces the investigation of the module space geometrically finite hyperbolic metrics on a 3-manifold with incompressible edge on the study of the space module compliant structures . (Each geometrically finite group corresponds to the Riemann surface , where the discontinuity area is. This generalizes Bers' uniformization theorem for quasifuchs groups .) ${\displaystyle M}$ ${\displaystyle \partial M}$ ${\displaystyle \partial M}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma \backslash \Omega }$ ${\displaystyle \Omega \subset \partial _{\infty }H^{3}}$

The Riemann surface corresponding to a geometrically finite group is called its conformal boundary . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma \backslash \Omega }$

literature

• 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
• Bowditch, BH: Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113 (1993) no. 2: 245-317.
• Bowditch, BH: Geometrical finiteness with variable negative curvature. Duke Math. J. 77 (1995) no. 1, 229-274.
• Bowditch, BH: Relatively hyperbolic groups. Boarding school J. Algebra Comput. 22 (2012), no. 3, 1250016, 66 pp.

Individual evidence

1. ↑ For the proof of equivalence see Theorem 3.7 in Matsuzaki-Taniguchi (op.cit.).
2. ^ Lipman Bers : Uniformization, Moduli, and Kleinian groups. Bull. London Math. Soc. 4: 257-300 (1972).
3. ↑ For the proof of equivalence see Bowditch (1993).
4. ↑ denotes the discontinuity area of . ${\displaystyle \Omega (\Gamma )\subset \partial _{\infty }X}$ ${\displaystyle \Gamma }$
5. ^ Greenberg, Leon: Fundamental polygons for Fuchsian groups. J. Analyze Math. 18 1967 99-105
6. ↑ M. Kapovich , L. Potyagailo : On the absence of Ahlfors' finiteness theorem for Kleinian in dimension three , top. Appl. 40, 83-91, 1991.