A Symplectic Map between Hyperbolic and Complex Teichmüller Theory

Let S be a closed, orientable surface of genus at least 2. The space TH ×ML, where TH is the “hyperbolic” Teichmüller space of S and ML is the space of measured geodesic laminations on S, is naturally a real symplectic manifold. The space CP of complex projective structures on S is a complex symplectic manifold. A relation between these spaces is provided by Thurston’s grafting map Gr. We prove that this map, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends.