Английская Википедия:Double limit theorem

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Шаблон:Short description In hyperbolic geometry, Thurston's double limit theorem gives condition for a sequence of quasi-Fuchsian groups to have a convergent subsequence. It was introduced in Шаблон:Harvtxt and is a major step in Thurston's proof of the hyperbolization theorem for the case of manifolds that fiber over the circle.

Statement

By Bers's theorem, quasi-Fuchsian groups (of some fixed genus) are parameterized by points in T×T, where T is Teichmüller space of the same genus. Suppose that there is a sequence of quasi-Fuchsian groups corresponding to points (gi, hi) in T×T. Also suppose that the sequences gi, hi converge to points μ,μШаблон:Prime in the Thurston boundary of Teichmüller space of projective measured laminations. If the points μ,μШаблон:Prime 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


Шаблон:Hyperbolic-geometry-stub