Grafting, pruning, and the antipodal map on measured laminations

Abstract

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmuller 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 $X \in \T(S)$, pruning $X$ gives a map $\ML(S) \to \T(S)$. 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 $\CP^1$ structures on $S$. For each $X \in \T(S)$, we show that the boundary of the space $P(X)$ of $\CP^1$ structures on $X$ in the compactification of the grafting coordinates is the graph $\Gamma(i_X)$ of the antipodal involution $i_X : \PML(S) \to \PML(S)$.