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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
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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
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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 (
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A geometrically finite polyhedron has only a finite number of cusps, and all but finitely many sides meet one of the cusps.
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showed that there are examples of finitely generated discrete groups in dimension 3 that are not geometrically finite.
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In hyperbolic spaces of dimension at most 2, finitely generated discrete groups are geometrically finite, but
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under this projection is a geometrically finite polyhedron with an infinite number of sides.
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that is convex, geometrically finite, and exact (every face is the intersection of
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in the conformal compactification of hyperbolic space has the following property:
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Greenberg, L. (1966), "Fundamental polyhedra for kleinian groups",
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in hyperbolic space is called geometrically finite if its closure
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with an infinite number of sides. The upper half plane model of
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if it can be described in terms of geometrically finite
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173:of isometries of hyperbolic space is called
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304:Foundations of hyperbolic manifolds
242:Density theorem for Kleinian groups
146:+1 dimensional hyperbolic space in
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224:A hyperbolic manifold is called
177:if it has a fundamental domain
220:Geometrically finite manifolds
63:Geometrically finite polyhedra
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300:Ratcliffe, John G. (1994),
165:Geometrically finite groups
154:, and the inverse image of
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138:≥2 there is a polyhedron
95:, there is a neighborhood
18:Geometrically finite group
43:if it has a well-behaved
103:such that all faces of
257:Annals of Mathematics
308:, Berlin, New York:
226:geometrically finite
175:geometrically finite
53:geometrically finite
41:geometrically finite
336:Hyperbolic geometry
209:, theorem 12.4.6).
125:For example, every
113:also pass through
45:fundamental domain
319:978-0-387-94348-0
260:, Second Series,
169:A discrete group
16:(Redirected from
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214:Greenberg (1966)
130:Euclidean space
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47:. A hyperbolic
37:hyperbolic space
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341:Kleinian groups
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310:Springer-Verlag
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270:10.2307/1970456
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264:: 433–441,
71:polyhedron
330:Categories
248:References
127:polyhedron
51:is called
39:is called
33:isometries
278:0003-486X
232:, 12.7).
201:, 12.4).
189:for some
236:See also
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109:meeting
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29:geometry
294:0200446
286:1970456
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282:JSTOR
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