In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain.
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 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) (Ratcliffe 1994, 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 (Ratcliffe 1994, theorem 12.4.6).