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Hyperbolic 3-manifolds and the geometry of the curve complex. (English) Zbl 1081.53032
Laptev, Ari (ed.), Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27–July 2, 2004. Zürich: European Mathematical Society (EMS) (ISBN 3-03719-009-4/hbk). 103-115 (2005).
Thurston’s ending lamination conjecture states that a tame hyperbolic 3-manifod $$M$$ is determined up to isometry by its topology and a certain finite set of end invariants. A proof of the ending lamination conjecture has been announced by J. F. Brock, R. D. Canary and Y. N. Minsky. The paper under review gives a survey of recent work towards and around the solution of Thurston’s ending lamination conjecture. Applications to the theory of surfaces and mapping class groups are given.
For the entire collection see [Zbl 1064.00004].

##### MSC:
 53C20 Global Riemannian geometry, including pinching 20F67 Hyperbolic groups and nonpositively curved groups 51M10 Hyperbolic and elliptic geometries (general) and generalizations 53C70 Direct methods ($$G$$-spaces of Busemann, etc.)
##### Keywords:
hyperbolic space; quasi-isometry; lamination conjecture