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Iteration on Teichmüller space. (English) Zbl 0695.57012
This paper contains the first published proof of a theorem of Thurston about hyperbolic structures on 3-manifolds. This theorem constitutes a key step in the proof of the Thurston’s geometrization theorem and deals with the following situation. Let M be a 3-manifold with incompressible boundary, and let \(\tau\) : \(\partial M\to \partial M\) be an orientation- reversing involution. Let M/\(\tau\) be the result of glueing M with itself by \(\tau\). Suppose that M has a geometrically finite hyperbolic structure. Then the theorem asserts that M/\(\tau\) has a hyperbolic structure iff M/\(\tau\) is a-toroidal.
This theorem can be reduced to the existence of a fixed point of the so- called skinning map of the Teichmüller space of \(\partial M\) associated with M and \(\tau\). (This reduction is due to Thurston himself.) The author proves the existence of such a fixed point by studying iterations of this map and applying the results of his previous paper “Amenability, Poincaré series and quasiconformal maps” [Invent. Math. 97, 95-127 (1989; Zbl 0672.30017)].
Reviewer: N.V.Ivanov

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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References:
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