Domains of discontinuity for almost-Fuchsian groups

Sanders, Andrew

doi:10.1090/tran/6789

Domains of discontinuity for almost-Fuchsian groups
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Trans. Amer. Math. Soc. 369 (2017), 1291-1308 Request permission

Abstract:

An almost-Fuchsian group $\Gamma <\mathrm {Isom}^{+}(\mathbb {H}^3)$ is a quasi-Fuchsian group such that the quotient hyperbolic manifold $\mathbb {H}^3/\Gamma$ contains a closed incompressible minimal surface with principal curvatures contained in $(-1,1).$ We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius $R>0$ in $\mathbb {C}\cup \{\infty \} =\partial _{\infty }(\mathbb {H}^3).$ This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal geometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups.

References

K. Astala and F. W. Gehring, Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood, Michigan Math. J. 32 (1985), no. 1, 99–107. MR 777305, DOI 10.1307/mmj/1029003136
Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2) 176 (2012), no. 1, 1–149. MR 2925381, DOI 10.4007/annals.2012.176.1.1
Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92. MR 903850
Charles L. Epstein, Envelopes of horospheres and weingarten surfaces in hyperbolic 3-space., Unpublished, 1984.
Charles L. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986), 96–135. MR 863521, DOI 10.1515/crll.1986.372.96
Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
Ren Guo, Zheng Huang, and Biao Wang, Quasi-Fuchsian 3-manifolds and metrics on Teichmüller space, Asian J. Math. 14 (2010), no. 2, 243–256. MR 2746123, DOI 10.4310/AJM.2010.v14.n2.a5
Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983. Notes taken by Peter Lax and John Gray; With a preface by S. S. Chern. MR 707850, DOI 10.1007/978-3-662-21563-0
Zheng Huang and Biao Wang, On almost-Fuchsian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 9, 4679–4698. MR 3066768, DOI 10.1090/S0002-9947-2013-05749-2
Troels Jørgensen, Compact $3$-manifolds of constant negative curvature fibering over the circle, Ann. of Math. (2) 106 (1977), no. 1, 61–72. MR 450546, DOI 10.2307/1971158
Yair Minsky, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2) 171 (2010), no. 1, 1–107. MR 2630036, DOI 10.4007/annals.2010.171.1
J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639–652. MR 654854, DOI 10.1090/S0002-9947-1982-0654854-8
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
William Thurston, Hyperbolic structures on 3-manifolds, II: surface groups and manifolds which fiber over the circle, Preprint, arXiv:math.GT/9801045.
Karen K. Uhlenbeck, Closed minimal surfaces in hyperbolic $3$-manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 147–168. MR 795233