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Boundaries of Teichmüller spaces and end-invariants for hyperbolic 3-manifolds. (English) Zbl 1011.30042
In boundaries for the Teichmüller space due to Bers and Thurston, geodesic laminations arise in natural ways. A point $$M$$ in Bers’s boundary, a hyperbolic 3-manifold, has an associated geodesic lamination $$\mathcal E(M)$$ that is pinched. The lamination $$\mathcal E(M)$$ is an invariant of the quasi-isometry class $$[M]$$ of $$M$$. A point $$[\mu]$$ in Thurston’s boundary, a measured lamination $$\mu$$ up to scale, records the asymptotic stretching of divergent hyperbolic metrics $$X_t\to [\mu]$$. Its support $$|\mu|$$ is a geodesic lamination. Thurston’s ending lamination conjecture predicts that the map $$[M]\mapsto\mathcal E(M)$$ from quasi-isometry classes in Bers’s boundary to the quotient of Thurston’s boundary by forgetting the measure is an injection. In other words, if one knows the lamination $$\mathcal E(M)$$, one knows the manifold $$M$$ up to quasi-isometry. The map $$\mathcal E$$ gives a bijection between dense subsets; the dense family of maximal cusps $$M$$ is mapped by $$\mathcal E$$ to the dense set of maximal partitions of $$S$$ by simple closed curves. Thus, given Thurston’s conjecture, it is natural to ask whether $$\mathcal E$$ is a homeomorphism. Or, how do sequences $$\mathcal E(M_n)$$ behave under limits $$M_n\to M$$? Let $$S$$ be an oriented compact topological surface of negative Euler characteristic with nonempty boundary. The author shows that $$\mathcal E$$ has the following continuity properties: the map $$\mathcal E$$ is a surjection onto the subset of laminations that relatively fill $$S$$; when $$\text{dim}_\mathbb C(\text{Teich}(S))>1$$, $$\mathcal E$$ is strictly lower-semicontinuous in the quotient topologies; $$\mathcal E$$ is continuous in a new end-invariant topology, based on the Hausdorff topology, which predicts new information about its limiting values; $$\mathcal E$$ cannot have a continuous inverse in the end-invariant topology, nor do Hausdorff limits completely encode the limiting end-invariant in general.

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) 57M50 General geometric structures on low-dimensional manifolds
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