“Geometric finiteness”的意思、由来-开放百科全书

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Geometric finiteness

释义

Geometrically finite polyhedra
Geometrically finite groups
Geometrically finite manifolds
References

In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite groups.

Geometrically finite polyhedra

A convex polyhedron C in hyperbolic space is called geometrically finite if its closure {{overline|C}} in the conformal compactification of hyperbolic space has the following property:

For each point x in {{overline|C}}, there is a neighborhood U such that all faces of {{overline|C}} meeting U also pass through x {{harv|Ratcliffe|1994|loc=12.4}}.

For example, every polyhedron with a finite number of faces is geometrically finite. In hyperbolic space of dimension at most 2, every geometrically finite polyhedron has a finite number of sides, but there are geometrically finite polyhedra in dimensions 3 and above with infinitely many sides. For example, in Euclidean space Rⁿ of dimension n≥2 there is a polyhedron P with an infinite number of sides. The upper half plane model of n+1 dimensional hyperbolic space in Rⁿ⁺¹ projects to Rⁿ, and the inverse image of P under this projection is a geometrically finite polyhedron with an infinite number of sides.

A geometrically finite polyhedron has only a finite number of cusps, and all but finitely many sides meet one of the cusps.

Geometrically finite groups

A discrete group G of isometries of hyperbolic space is called geometrically finite if it has a fundamental domain C that is convex, geometrically finite, and exact (every face is the intersection of C and gC for some g ∈ G) {{harv|Ratcliffe|1994|loc=12.4}}.

In hyperbolic spaces of dimension at most 3, every exact, convex, fundamental polyhedron for a geometrically finite group has only a finite number of sides, but in dimensions 4 and above there are examples with an infinite number of sides {{harv|Ratcliffe|1994|loc=theorem 12.4.6}}.

In hyperbolic spaces of dimension at most 2, finitely generated discrete groups are geometrically finite, but {{harvtxt|Greenberg|1966}} showed that there are examples of finitely generated discrete groups in dimension 3 that are not geometrically finite.

Geometrically finite manifolds

A hyperbolic manifold is called geometrically finite if it has a finite number of components, each of which is the quotient of hyperbolic space by a geometrically finite discrete group of isometries {{harv|Ratcliffe|1994|loc=12.7}}.

References

{{Citation | last1=Greenberg | first1=L. | title=Fundamental polyhedra for kleinian groups | jstor=1970456 | mr=0200446 | doi=10.2307/1970456 | year=1966 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=84 | pages=433–441}}
{{Citation | last1=Ratcliffe | first1=John G. | title=Foundations of hyperbolic manifolds | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-94348-0 | year=1994}}

2 : Hyperbolic geometry|Kleinian groups

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