# zbMATH — the first resource for mathematics

Limits of quasi-Fuchsian groups with small bending. (English) Zbl 1081.30038
The author studies the limits of quasi-Fuchsian groups for which the bending measures on the convex hull boundary tend to zero, and several necessary and sufficient conditions for the existence and being Fuchsian of the limit group are given. As a consequence, it is proved that the answer to Conjecture $$6.5$$ in [Geom. Dedicata 88, 211–237 (2001; Zbl 1005.30032)] raised by the author is positive. A large class of cone manifolds which degenerate to hyperbolic surfaces as the cone angles approach $$2\pi$$ has been obtained by doubling the author’s examples.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
Full Text:
##### References:
 [1] A. F. Beardon, The Geometry of Discrete Groups , Grad. Texts in Math. 91 , Springer, New York, 1995. [2] F. Bonahon and J.-P. Otal, Laminations mesurées de plissage des variétés hyperboliques de dimension 3 , to appear in Ann. of Math. (2). JSTOR: · Zbl 1083.57023 [3] R. Brooks and J. P. Matelski, Collars in Kleinian groups , Duke Math. J. 49 (1982), 163–182. · Zbl 0484.30029 [4] R. D. Canary, D. B. A. Epstein, and P. Green, “Notes on notes of Thurston” in Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, U.K., 1984) , ed. D. B. A. Epstein, London Math. Soc. Lecture Note Ser. 111 , Cambridge Univ. Press, Cambridge, 1987, 3–92. [5] R. Díaz and C. Series, Examples of pleating varieties for twice punctured tori , Trans. Amer. Math. Soc. 356 (2004), no. 2, 621–658. · Zbl 1088.30043 [6] –. –. –. –., Limit points of lines of minima in Thurston’s boundary of Teichmüller space , Algebr. Geom. Topol. 3 (2003), 207–234. · Zbl 1066.32020 [7] D. B. A. Epstein and A. Marden, “Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces” in Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, U.K., 1984) , ed. D. B. A. Epstein, London Math. Soc. Lecture Note Ser. 111 , Cambridge Univ. Press, Cambridge, 1987, 113–253. · Zbl 0612.57010 [8] L. Keen and C. Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori , Topology 32 (1993), 719–749. · Zbl 0794.30037 [9] –. –. –. –., Continuity of convex hull boundaries , Pacific J. Math. 168 (1995), 183–206. · Zbl 0838.30043 [10] –. –. –. –., “How to bend pairs of punctured tori” in Lipa’s Legacy (New York, 1995) , ed. J. Dodziuk and L. Keen, Contemp. Math. 211 , Amer. Math. Soc., Providence, 1997, 359–388. [11] –. –. –. –., Pleating invariants for punctured torus groups , Topology 43 (2004), 447–491. · Zbl 1134.30030 [12] S. P. Kerckhoff, The Nielsen realization problem , Ann. of Math. (2) 117 (1983), 235–265. JSTOR: · Zbl 0528.57008 [13] –. –. –. –., Earthquakes are analytic , Comment. Math. Helv. 60 (1985), 17–30. · Zbl 0575.32024 [14] –. –. –. –., Lines of minima in Teichmüller space , Duke Math. J. 65 (1992), 187–213. · Zbl 0771.30043 [15] C. Kourouniotis, Complex length coordinates for quasi-Fuchsian groups , Mathematika 41 (1994), 173–188. · Zbl 0801.30036 [16] C. Lecuire, Plissage des variétés hyperboliques de dimension $$3$$ , available from http://www.maths.warwick.ac.uk/ clecuire A. Marden, The geometry of finitely generated Kleinian groups , Ann. of Math. (2) 99 (1974), 383–462. JSTOR: · Zbl 0282.30014 [17] C. T. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics, III: The Teichmüller space of a holomorphic dynamical system , Adv. Math. 135 (1998), 351–395. · Zbl 0926.30028 [18] Y. N. Minsky, Harmonic maps into hyperbolic $$3$$-manifolds , Trans. Amer. Math. Soc. 332 (1992), 607–632. · Zbl 0762.53040 [19] D. Mumford, C. Series, and D. Wright, Indra’s Pearls: The Vision of Felix Klein , Cambridge Univ. Press, New York, 2002. · Zbl 1141.00002 [20] J.-P. Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension $$3$$ , Astérisque 235 , Soc. Math. France, Paris, 1996. · Zbl 0855.57003 [21] J. R. Parker and C. Series, Bending formulae for convex hull boundaries , J. Anal. Math. 67 (1995), 165–198. · Zbl 0849.30036 [22] R. C. Penner and J. L. Harer, Combinatorics of Train Tracks . Ann. of Math. Stud. 125 , Princeton Univ. Press, Princeton, 1992. · Zbl 0765.57001 [23] C. Series, On Kerckhoff minima and pleating loci for quasi-Fuchsian groups , Geom. Dedicata 88 (2001), 211–237. · Zbl 1005.30032 [24] D. Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension $$3$$ fibrées sur $$S^1$$ , Séminaire Bourbaki 1979/80, no. 554, Lecture Notes in Math. 842 , Springer, Berlin, 1981, 196–214. · Zbl 0459.57006 [25] W. P. Thurston, Hyperbolic structures on $$3$$-manifolds, II: Surface groups and $$3$$-manifolds which fiber over the circle , [26] ——–, Geometry and topology of three-manifolds , Princeton University lecture notes, 1980, available from http://www.msri.org/publications/books/gt3m/ [27] S. Wolpert, The Fenchel-Nielsen deformation , Ann. of Math. (2) 115 (1982), 501–528. JSTOR: · Zbl 0496.30039 [28] –. –. –. –., On the symplectic geometry of deformations of a hyperbolic surface , Ann. of Math. (2) 117 (1983), 207–234. JSTOR: · Zbl 0518.30040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.