On Thurston’s boundary of Teichmüller space and the extension of earthquakes. (English) Zbl 0761.57009
A celebrated result of W. Thurston describes a boundary attached to the Teichmüller space of a finite-area hyperbolic surface. Its points are measured laminations on the surface (up to scalar multiple), and limiting toward a boundary point is understood in terms of intersection number with the measured lamination: the limiting relative lengths of loops in the hyperbolic structures are given by the relative values of their intersection with the measured lamination. Measured laminations are closely related to measured foliations, which provide alternative representatives for the points of the Thurston boundary.
In this paper, the author gives a new criterion for the convergence of a sequence of hyperbolic metrics to a point on Thurston’s boundary. Thurston gave a global parameterization of Teichmüller space which takes a hyperbolic structure to an equivalence class of measured foliations transverse to a fixed geodesic lamination (whose complementary regions are ideal triangles). The key technical result of the paper provides inequalities relating the lengths of simple closed geodesics for a hyperbolic metric to the intersection numbers of their isotopy classes with the measured foliations associated to the metric. This enables the parameterization to be extended to the boundary, and the earthquake flow, suitably normalized, to be extended to the boundary. This extension of this earthquake flow has also been discovered by F. Bonahon.
In this paper, the author gives a new criterion for the convergence of a sequence of hyperbolic metrics to a point on Thurston’s boundary. Thurston gave a global parameterization of Teichmüller space which takes a hyperbolic structure to an equivalence class of measured foliations transverse to a fixed geodesic lamination (whose complementary regions are ideal triangles). The key technical result of the paper provides inequalities relating the lengths of simple closed geodesics for a hyperbolic metric to the intersection numbers of their isotopy classes with the measured foliations associated to the metric. This enables the parameterization to be extended to the boundary, and the earthquake flow, suitably normalized, to be extended to the boundary. This extension of this earthquake flow has also been discovered by F. Bonahon.
Reviewer: D.McCullough (Norman)
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
| 30F60 | Teichmüller theory for Riemann surfaces |
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
Keywords:
Teichmüller space; convergence of a sequence of hyperbolic metrics to a point on Thurston’s boundary; geodesic lamination; lengths of simple closed geodesics; intersection numbers; earthquake flow; measured foliationsReferences:
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