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Hyperbolic structures on 3-manifolds. I: Deformation of acylindrical manifolds. (English) Zbl 0668.57015
This is the first in a long-awaited series of papers in which the author plans to give the proof of his theorem that the interior of an atoroidal Haken 3-manifold admits a complete hyperbolic structure. At the end of the series, he also plans to give the proof that his Geometrisation conjecture holds for compact irreducible 3-orbifolds with a singular set which is neither empty nor 0-dimensional.
Let M be a Haken 3-manifold. Then M contains an incompressible surface F and cutting M along F yields a Haken manifold \(M_ 1\) (which may not be connected). One can repeat this cutting procedure to eventually obtain \(M_ n\), which is a disjoint union of 3-balls. The minimum value of n taken over all such cutting sequences is called the length of M. The author’s hyperbolisation result for Haken 3-manifolds is proved by induction on the length, as are most results about Haken manifolds. For the induction step, one has a Haken manifold \(M_ 1\), whose interior admits a complete hyperbolic structure, and one wants to show that the manifold M obtained from \(M_ 1\) by glueing two copies of \(F_ 1\) in \(\partial M_ 1\) also admits a hyperbolic structure. The property that M is atoroidal is crucial here. The argument consists of analysing the space of deformations of the hyperbolic structure on \(M_ 1\), called \(AH(M_ 1)\), and showing that there is a deformation such that one can glue the new structure to obtain a hyperbolic structure on M. The existence of such a deformation is equivalent to the existence of a fixed point for a certain map from \(AH(M_ 1)\) to itself. If \(AH(M_ 1)\) is compact, it is less difficult to show that the required fixed point exists.
In the paper under review, the author shows that if M is an acyclindrical 3-manifold then AH(M) is compact, where acylindrical means that \(\partial M\) is incompressible and that any proper map of the annulus A to M, which injects \(\pi_ 1(A)\), is properly homotopic into \(\partial M\). This is a generalization of Mostow’s rigidity theorem, which asserts that if M has finite volume, than AH(M) consists of at most one point. The author’s arguments are basically geometric. He uses the uniform injectivity of pleated surfaces, a result which he proves after giving a brief introduction to the theory of pleated surfaces.
There is also a largely algebraic proof that AH(M) is compact due to J. W. Morgan and P. Shalen [ibid. 120, 410-476 (1984; Zbl 0583.57005)].

57N10 Topology of general \(3\)-manifolds (MSC2010)
58D17 Manifolds of metrics (especially Riemannian)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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