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Shrinkwrapping and the taming of hyperbolic 3-manifolds. (English) Zbl 1090.57010
The classification of hyperbolic structures on $$3$$-manifolds has undergone vigorous investigation during the past three decades, which reduced several of its most important conjectures to the problem of understanding the topological ends of hyperbolic $$3$$-manifolds with finitely generated fundamental group. A key remaining issue was Marden’s Tameness Conjecture, which says that such an end must be topologically the product of a surface and a half-open interval, or equivalently that the hyperbolic $$3$$-manifold is the interior of a compact $$3$$-manifold with boundary. This was finally proven independently by I. Agol [Tameness of hyperbolic $$3$$-manifolds, ArXiv GT/0405568] and by the authors, in work which constitutes the paper under review. In combination with earlier work of numerous investigators, this establishes the Ahlfors Measure Conjecture, the Bers Density Conjecture, and the Classification Theorem, which says that a hyperbolic $$3$$-manifold with finitely generated fundamental group is determined up to isometry by its topological type and its end data, the latter consisting of the conformal boundary of its geometrically finite ends and the ending laminations of its geometrically infinite ends.
A major case of the tameness question was resolved by F. Bonahon. Building upon ideas of Thurston, he proved that ends of hyperbolic $$3$$-manifolds are tame provided that they contain a sequence of surfaces satisfying certain geometric conditions, which nowadays are stated in terms of a $$\text{CAT}(-1)$$-condition, and moreover proved that such a sequence exists provided that the end satisfies certain incompressibility conditions. In the recent work of Agol and the authors, the substantial remaining technical difficulties of the general case have been overcome.
The paper draws upon a remarkable assemblage of geometric and topological methodology. The shrinkwrapping technique which gives the paper its title is a method for producing surfaces in an end of a hyperbolic $$3$$-manifold. Roughly speaking, a shrinkwrapped surface is a minimal area surface in $$M$$ which misses a given finite union $$\Gamma$$ of disjoint simple closed geodesics and has minimal area among such surfaces. A key step involves finding a homotopy in $$M-\Gamma$$ carrying a surface in $$M-\Gamma$$ to a shrinkwrapped surface. Also noteworthy is the use of a generalized version of topological ideas of M. Brin and T. Thickstun to produce a suitable sequence of surfaces to use for the shrinkwrapping.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 49Q10 Optimization of shapes other than minimal surfaces 57N10 Topology of general $$3$$-manifolds (MSC2010)
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