geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a group of

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...

is called geometrically finite if it has a well-behaved

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

. A hyperbolic

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

is called geometrically finite if it can be described in terms of geometrically finite

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

Geometrically finite polyhedra

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

polyhedron ''C'' in hyperbolic space is called geometrically finite if its closure in the conformal compactification of hyperbolic space has the following property: *For each point ''x'' in , there is a neighborhood ''U'' of ''x'' such that all faces of meeting ''U'' also pass through ''x'' . For example, every

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

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 R^''n'' 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 R^''n''+1 projects to R^''n'', 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'') . 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.

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 .

References

* *{{Citation , last1=Ratcliffe , first1=John G. , title=Foundations of hyperbolic manifolds , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , isbn=978-0-387-94348-0 , year=1994 , url-access=registration , url=https://archive.org/details/foundationsofhyp0000ratc Hyperbolic geometry Kleinian groups

Geometrically finite polyhedra

Geometrically finite groups

Geometrically finite manifolds

See also

References