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End invariants and the classification of hyperbolic 3-manifolds. (English) Zbl 1049.57010
Jerison, David (ed.) et al., Current developments in mathematics, 2002. Proceedings of the joint seminar by MIT and Harvard, Cambridge, MA, 2002. Somerville, MA: International Press (ISBN 1-57146-102-7/hbk). 181-217 (2003).
From the text: “These notes are a biased guide to some recent developments in the deformation theory of hyperbolic 3-manifolds and Kleinian groups.” “We will focus on the geometric study of ends of hyperbolic 3-manifolds and boundaries of deformation spaces, and in particular on the techniques that led to the recent solution by Brock, Canary and the author of the incompressible-boundary case of Thurston’s Ending Lamination Conjecture” (which roughly states that a hyperbolic 3-manifold is determined by its end invariants: It is an element of a Teichm├╝ller space of a surface if one end is geometrically finite, or it is an ending lamination if one end is simply degenerate; Bonahon proved that in the incompressible-boundary case every end is geometrically finite, that is of one of these two types). The paper gives also an introduction to the history and literature of the subject.
For the entire collection see [Zbl 1033.00012].

MSC:
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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