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Hyperbolic surface bundles over the circle. (English) Zbl 0959.57019

Boileau, Michel (ed.) et al., Progress in knot theory and related topics. Paris: Hermann. Trav. Cours. 56, 121-142 (1997).
From the introduction: In the class of geometric 3-manifolds, by far the most inaccessible and difficult are the hyperbolic 3-manifolds. In this paper, we study a restricted yet rich subclass, namely those which are surface bundles over the circle. A remarkable theorem of W. P. Thurston [Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fibre over the circle, Preprint] asserts that such a bundle admits a hyperbolic structure if and only if the associated monodromy map is isotopic to a pseudo-Anosov map of the fibre. In principle at least, this reduces the understanding of these manifolds to certain questions about surface automorphisms. In what follows we shall restrict attention to those bundles which fibre in infinitely many different ways. Some of the results in this paper are known to researchers in dynamical systems. In particular, D. Fried has written several papers on the same subject [Comment. Math. Helv. 57, 237-259 (1982; Zbl 0503.58026); Astérisque 66-67, 251-266 (1979; Zbl 0446.57023)]. The main novelty of our approach is that we work largely from the topological point of view. A portion of this paper is devoted to proving some of Fried’s results by different methods; though we obtain estimates which are not contained in his papers. Since the original writing of this paper, there has been much further work done on one of our main tools, namely essential laminations [D. Gabai and U. Oertel, Ann. Math. (2), 130, No. 1, 41-73 (1989; Zbl 0685.57007) and S. Matsumoto, J. Fac. Sci., Univ. Tokyo, Sect. IA 34, 763-778 (1987; Zbl 0647.57006)].
For the entire collection see [Zbl 0913.00020].

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57R22 Topology of vector bundles and fiber bundles
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