## 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

Zbl 0672.30017
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### References:

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