hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...

, Thurston's double limit theorem gives condition for a sequence of

quasi-Fuchsian group In the mathematical theory of Kleinian groups, a quasi-Fuchsian group is a Kleinian group whose limit set is contained in an invariant Jordan curve. If the limit set is equal to the Jordan curve the quasi-Fuchsian group is said to be of type one, a ...

s to have a convergent subsequence. It was introduced in and is a major step in Thurston's proof of the

hyperbolization theorem In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. Statement One form of Thurston's geometrization theor ...

for the case of manifolds that fiber over the circle.

Statement

By Bers's theorem,

s (of some 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 nom ...

) are parameterized by points in ''T''×''T'', where ''T'' is

Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...

of the same genus. Suppose that there is a sequence of quasi-Fuchsian groups corresponding to points (''g''_''i'', ''h''_''i'') in ''T''×''T''. Also suppose that the sequences ''g''_''i'', ''h''_''i'' converge to points μ,μ in the Thurston boundary of Teichmüller space of projective measured laminations. If the points μ,μ have the property that any nonzero measured lamination has positive intersection number with at least one of them, then the sequence of quasi-Fuchsian groups has a subsequence that converges algebraically.

References

* * * Translated into English as * Kleinian groups Hyperbolic geometry Theorems in geometry {{hyperbolic-geometry-stub