Notes on Tameness
The following theorem is generally referred to as the ``Tameness Theorem''. Let \(M\) be a complete hyperbolic 3-manifold with \(\pi_1(M)\) finitely generated. Then \(M\) is homeomorphic to the interior of a compact manifold with boundary. This was proven in the ``indecomposable'' case by \textit{F. Bonahon} [Ann. Math. (2) 124, 71--158 (1986; Zbl 0671.57008)] and in general independently by \textit{I. Agol} [Tameness and hyperbolic 3-manifolds, Preprint, Chicago (2004)] and \textit{D. Calegari} and \textit{D. Gabai} [J. Am. Math. Soc. 19, No.~2, 385-446 (2006; Zbl 1090.57010)]. The paper under review is about a general version of the ``Tameness Theorem'' and Ahlfors' finiteness theorem which the author presents in the following form. Let \(M\) be a non-elementary (this means \(\pi_1(M)\) is not trivial, \(\mathbb{Z}\) or \(\mathbb{Z}\oplus\mathbb{Z}\)) complete hyperbolic 3-manifold with \(\pi_1(M)\) finitely generated. Let \(t> 0\), and let \(F_t\) be the boundary of the \(t\)-neighbourhood of the convex core of \(M\). Then \(F_t\) has finite area. Here by ``convex core'' of \(M\) the author understands essentially (it means not degenerate to a totally geodesic surface) the unique smallest submanifold which carries the whole of \(\pi_1(M)\) and has convex boundary. Moreover the metric \(t\)-neighbourhood of the convex core is a \(C^1\)-submanifold. \(F_t\) is the boundary of this neighbourhood and it is an embedded \(C^1\)-surface (possible empty or disconnected). The above mentioned generalization is the following. Let \(M\) be a complete Riemannian 3-manifold of pinched negative curvature with \(\pi_1(M)\) finitely generated. Then, 1. \(M\) is homeomorphic to the interior of a compact manifold with boundary; 2. \(F_t\) has finite area. The paper has 8 chapters. Let us present their titles from the second to the eighth: Topological constructions; Polyhedral constructions; The wrapping construction; Proof of tameness when there are no cusps; Tameness with cusps; Variable curvature; Ahlfors' finiteness theorem.
interior of a compact manifold with boundary
scientific article; zbMATH DE number 5896744