## Iteration of mapping classes and limits of hyperbolic 3-manifolds.(English)Zbl 0969.57011

This is a profound study of algebraic and geometric limits of sequences of quasi-Fuchsian (Kleinian) groups resp. of the associated hyperbolic and Kleinian 3-manifolds (homeomorphic to the (interior of the) product of a surface with a real interval), generated by iteration of an element of the mapping class group of the surface. More precisely, given a (closed) orientable surface $$S$$ of negative Euler characteristic, any two points $$X$$ and $$Y$$ in the Teichmüller space of $$S$$ can be simultaneously uniformized by a quasi-Fuchsian group $$\Gamma (X,Y)$$, with associated quasi-Fuchsian 3-manifold $$Q(X,Y) = \mathbb H^3/\Gamma (X,Y)$$ such that $$\partial Q(X,Y) = X \cup Y$$. Fixing $$Y$$ and letting vary $$X$$ one obtains, as a copy of the Teichmüller space of $$S$$, the Bers slice in the representation variety of $$\pi_1S$$ in the isometry group of hyperbolic 3-space $$\mathbb H^3$$ whose closure gives the Bers compactification of the Teichmüller space. The problem considered in the paper is the question of algebraic and geometric convergence of the quasi-Fuchsian 3-manifolds $$Q(\phi^iX,Y)$$, for a mapping class (element of the modular group) $$\phi$$ of the surface $$S$$, and of the behaviour of the surfaces $$\phi^iX$$ in a limit (creation of new parabolic elements, geometrically finite or degenerate ends). If $$\phi$$ is pseudo-Anosov, convergence was known by results of Cannon and Thurston; the surfaces $$\phi^iX$$ degenerate, leaving a totally degenerate limit $$Q_{\phi}$$ of the sequence of quasi-Fuchsian 3-manifolds (i.e. $$\partial Q_{\phi} = Y$$, with no new parabolic elements, and the algebraic and geometric limits coincide (“strong convergence”). In the present paper, the case of a general mapping class $$\phi$$ is studied.
The main result of the paper is the following. There is an essential subsurface $$D$$ associated to $$\phi$$ such that any geometric limit of the collection $$Q(\phi^iX,Y)$$, $$i \in \mathbb N$$, has homeomorphism type $$S \times \mathbb R - D \times 0$$ if $$D \neq S$$, and $$S \times \mathbb R$$ otherwise. The complement $$S - D$$ is the maximal subsurface on which $$\phi$$ restricts to a finite order mapping class. The components of $$D$$ corresponding to pseudo-Anosov restrictions of $$\phi$$ give new simply degenerate ends of the geometric limit. It is also shown that there exists a positive integer $$s$$ such that the sequence $$Q(\phi^{si}X,Y)$$ converges algebraically and geometrically, and explicit quasi-isometric models for the limits are given.

### MSC:

 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57N10 Topology of general $$3$$-manifolds (MSC2010)
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