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Homotopy hyperbolic 3-manifolds are hyperbolic. (English) Zbl 1052.57019

Using a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds, the authors complete the proof of several long-standing conjectures in 3-manifold theory. The main theorem of the article is the following fundamental result. Let \(N\) be a closed hyperbolic 3-manifold. Then (i) if \(f : M \to N\) is a homotopy equivalence, where \(M\) is a closed irreducible 3-manifold, then \(f\) is homotopic to a homeomorphism; (ii) if \(f, g : M \to N\) are homotopic homeomorphisms, then \(f\) is isotopic to \(g\); (iii) the space of hyperbolic metrics on \(N\) is path connected. Remark that under the additional hypothesis that \(M\) is hyperbolic, conclusion (i) follows from Mostow’s rigidity theorem.
Moreover, the authors proved the following volume lower bound. If \(N\) is a closed hyperbolic 3-manifold then the volume of \(N\) is greater than \(0.16668\ldots\). Remark that recently A. Przeworski improved the volume lower bound to \(0.3315\). (The smallest known orientable hyperbolic 3-manifold has volume \(0.94\ldots\).)

MSC:

57M50 General geometric structures on low-dimensional manifolds
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