Projective structures, grafting and measured laminations

Abstract

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmüller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective (CP 1 ) structures on a surface.We also study the rays in Teichmüller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.30F60; 30F10, 30F40, 32G15, 57M50

Bers slices are Zariski dense

“…The map λ → gr −1 λ (Y ) is injective (see [11]), establishing a bijection between W Y ∩ T(S) and ML Z (S). In particular, the set W Y ∩ T(S) is infinite, which is used in [10] to show that W Y is not an algebraic variety.…”

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

“…Grafting construction This is more widespread in the literature (see e.g. [16] or [9] for an account). The idea is to cut S open along γ and insert a Euclidean cylinder [0, l] × γ , l ≥ 0 with the appropriate twist parameter (details below (1.4)).…”

Complex projective structures

“…Building on the Scannell-Wolf result, the author and Wolf showed that the X-grafting map is also a local homeomorphism, leading to: Theorem 4.4 (Dumas and Wolf [27]). For each X ∈ T (S), the X-grafting map gr X : ML(S) → T (S) is a homeomorphism.…”