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.
A convex polyhedron C in hyperbolic space is called geometrically finite if its closure in the conformal compactification of hyperbolic space has the following property:
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.
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) .
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 .
In hyperbolic spaces of dimension at most 2, finitely generated discrete groups are geometrically finite, but showed that there are examples of finitely generated discrete groups in dimension 3 that are not geometrically finite.
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 .