Double limit theorem

Short description: On when a sequence of quasi-Fuchsian groups have a convergent subsequence

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 (Thurston 1998) 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 (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

• Holt, John (2001), The double limit theorem, http://www.cmp.caltech.edu/~dannyc/courses/geom/holt_notes.ps.gz, retrieved 2011-03-20 
• Kapovich, Michael (2009) [2001], Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8 
• Otal, Jean-Pierre (1996), "Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3", Astérisque (235), ISSN 0303-1179  Translated into English as Otal, Jean-Pierre (2001) [1996], Kay, Leslie D., ed., The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, 7, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2153-4, https://books.google.com/books?id=pVObtYVehxIC 
• Thurston, William P. (1998) [1986], Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle