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Bounded combinatorics and uniform models for hyperbolic 3-manifolds. (English) Zbl 1353.51014

The authors prove
Theorem: Let \(M\) be a finite collection of decorated manifolds and fix \(R>0\). There exist \(D\) and \(K\) such that, for any \(M\)-gluing \(X\) with \(R\)-bounded combinatorics and all heights greater that \(D\), \(X\) admits a unique hyperbolic metric \(\sigma\). Moreover, there exists a \(K\)-bilipschitz homeomorphism from the model \(M_X\) to \((X, \sigma)\) in the correct isotopy class.
This is an extension of the ending lamination theorem, which gives a relation between the geometric features of a hyperbolic 3-manifold to its topological description.

MSC:

51M10 Hyperbolic and elliptic geometries (general) and generalizations
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