×

zbMATH — the first resource for mathematics

Geometric inflexibility of hyperbolic cone-manifolds. (English) Zbl 1446.57017
Fujiwara, Koji (ed.) et al., Hyperbolic geometry and geometric group theory. Proceedings of the 7th Seasonal Institute of the Mathematical Society of Japan (MSJ-SI), Tokyo, Japan, July 30 – August 5, 2014. Tokyo: Mathematical Society of Japan (MSJ). Adv. Stud. Pure Math. 73, 47-64 (2017).
Summary: We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lipschitz constant decays exponentially in the distance from the cone-singularity. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves.
For the entire collection see [Zbl 1380.20002].
MSC:
57M50 General geometric structures on low-dimensional manifolds
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] M. Bridson and A. Haefliger.Metric Spaces of Non-Positive Curvature. Springer-Verlag, 1999. · Zbl 0988.53001
[2] J. Brock and K. Bromberg. On the density of geometrically finite Kleinian groups.Acta Math.192(2004), 33-93. · Zbl 1055.57020
[3] J. Brock and K. Bromberg. Geometric inflexibility and 3-manifolds that fiber over the circle.Journal of Topology4(2011), 1-38. · Zbl 1220.30057
[4] J. Brock, K. Bromberg, R. Evans, and J. Souto. Maximal cusps, ending laminations and the classification of Kleinian groups.In preparation (2008).
[5] K. Bromberg. Hyperbolic cone manifolds,short geodesics,and Schwarzian derivatives.J. Amer. Math. Soc.17(2004), 783-826. · Zbl 1061.30037
[6] K. Bromberg. Drilling long geodesics in hyperbolic 3-manifolds.Preprint (2006). · Zbl 1101.57006
[7] K. Bromberg. Projective structures with degenerate holonomy and the Bers density conjecture.Annals of Math.166(2007), 77-93. · Zbl 1137.30014
[8] K. Bromberg and J. Souto. Density of Kleinian groups.In preparation.
[9] R. Brooks and J. P. Matelski. Collars for Kleinian Groups.Duke Math. J.49(1982), 163-182. · Zbl 0484.30029
[10] C. Hodgson and S. Kerckhoff. Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery.J. Diff. Geom.48(1998), 1-59. · Zbl 0919.57009
[11] C. Hodgson and S. Kerckhoff. Universal bounds for hyperbolic Dehn surgery.Ann. Math.162(2005), 367-421. · Zbl 1087.57011
[12] C. Hodgson and S. Kerckhoff. The shape of hyperbolic Dehn surgery space.Geometry and Topology12(2008), 1033-1090. · Zbl 1144.57015
[13] Y. Kamishima and Ser P. Tan. Deformation spaces on geometric structures. In Y. Matsumoto and S. Morita, editors,Aspects of Low Dimensional Manifolds. Published for Math. Soc. of Japan by Kinokuniya Co., 1992. · Zbl 0798.53030
[14] S. Kojima. Deformations of hyperbolic 3-cone-manifolds.J. Differential Geom.49(1998), 469-516. · Zbl 0990.57004
[15] C. McMullen.Renormalization and 3-Manifolds Which Fiber Over the Circle. Annals of Math. Studies 142, Princeton University Press, 1996. · Zbl 0860.58002
[16] Y.
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.