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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)].

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 |

### Keywords:

hyperbolic structures on 3-manifolds; 3-manifold with incompressible boundary; orientation-reversing involution; skinning map of the Teichmüller space### Citations:

Zbl 0672.30017### References:

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