In mathematics, Mumford's compactness theorem states that the space of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
s of fixed
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
''g'' > 1 with no
closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic fl ...
s of length less than some fixed ''ε'' > 0 in the
Poincaré metric
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometr ...
is compact. It was proved by as a consequence of a theorem about the compactness of sets of discrete subgroups of
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s generalizing
Mahler's compactness theorem
In mathematics, Mahler's compactness theorem, proved by , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in whi ...
.
References
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Riemann surfaces
Kleinian groups
Compactness theorems
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