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Drilling short geodesics in hyperbolic 3-manifolds. (English) Zbl 1101.57006
Minsky, Yair (ed.) et al., Spaces of Kleinian groups. Proceedings of the programme ‘Spaces of Kleinian groups and hyperbolic 3-manifolds’, Cambridge, UK, July 21–August 15, 2003. Cambridge: Cambridge University Press (ISBN 0-521-61797-9/pbk). London Mathematical Society Lecture Note Series 329, 1-27 (2006).
The author reviews the deformation theory of finite volume 3-dimensional cone-manifolds as developed by Hodgson and Kerckhoff, with extensions due to the author himself to the context of infinite volume geometrically finite hyperbolic cone-manifolds. The term “drilling” in the title of the paper refers to an operation which decreases a cone angle. The author also presents recent theorems he obtained that allow the control of various geometric quantities (lengths of geodesics, change in the projective structure of the boundary, length of the singular locus) during a process of drilling. The author also surveys some relations of these deformation techniques to recent developments on some classical conjectures on Kleinian groups, namely, the density conjecture, the density of cusps on the boundary of quasiconformal deformation space, and the ending lamination conjecture.
For the entire collection see [Zbl 1089.30004].

57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57N10 Topology of general \(3\)-manifolds (MSC2010)