Boundaries of Teichmüller spaces and end-invariants for hyperbolic 3-manifolds. (English) Zbl 1011.30042

Brock, Jeffrey F.

Duke Math. J. 106, No. 3, 527-552 (2001).

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.

Reviewer: Vasily A.Chernecky (Odessa)

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

Keywords:

boundaries for the Teichmüller space; measured geodesic lamination; Thurston’s boundary; surjectivity onto measurable lamination; maximal cusps; Bers boundary; quotient topology; subsurfaces; pleated surfaces; lengths of laminations; decomposing laminations; pinching deformation; lower-semicontinuity; spinning maximal cusps; continuity in the end-invariant topology; convergence in a Bers compactification; constructing partitions; failure of Hausdorff topology; implicit cusps

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