Grafting, Pruning, and the Antipodal Map on Measured Laminations

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmüller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of grafting. Specifically, for each conformal structure X ∈ T (S), pruning X gives a map ML (S) → T (S). We show that this map extends to the Thurston compactification of T (S), and that its boundary values are the natural antipodal involution relative to X on the space of projective measured laminations. We use this result to study Thurston’s grafting coordinates on the space of CP structures on S. For each X ∈ T (S), we show that the boundary of the space P (X) of CP structures on X in the compactification of the grafting coordinates is the graph Γ(iX) of the antipodal involution iX : PML (S) → PML (S).