Grafting, pruning, and the antipodal map on measured laminations

Abstract

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 conformai 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 X space of projective measured laminations. We use this result to study Thurston’s grafting coordinates on the space of ℂℙ1 structures on S. For each X ∈ T(S), we show that the boundary of the space P (X) of ℂℙ1 structures on X in the compactification of the grafting coordinates is the graph Γ(iX) of the antipodal involution iX: ℙML(S) → ℙML(S). © 2006 Applied Probability Trust.