Fixed points of compositions of earthquakes

Abstract

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.

19 pages, 1 figure. v2: 21 pages, 3 figures. v2 is a substantial rewrite, with simpler proofs and better explanations, some corrections. v3: further improvements in the exposition

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• Kleinian groups which are almost fuchsian
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• Maximal surfaces and the universal Teichmüller space

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• The convex core of quasifuchsian manifolds with particles
• Regular globally hyperbolic maximal anti-de Sitter structures
• Bending laminations on convex hulls of anti-de Sitter quasicircles
• Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces
• On the Weyl problem for complete surfaces in the hyperbolic and anti-de Sitter spaces
• Group actions and scattering problems in Teichmüller theory
• Maximal surfaces and the universal Teichmüller space
• Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds
• Some open questions on anti-de Sitter geometry
• The Anti-de Sitter proof of Thurston's earthquake theorem
• Hamiltonian properties of earthquakes flows on surfaces with closed geodesic boundary
• On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds