Iteration of mapping classes and limits of hyperbolic 3-manifolds.

*(English)*Zbl 0969.57011This 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.

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.

Reviewer: Bruno Zimmermann (Trieste)