Grafting, Pruning, and the Antipodal Map on Measured Laminations

David Dumas

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 conformal structure X $ \in$ $ \mathscr$T(S), pruning X gives a map $ \mathscr$ML(S)$ \to$$ \mathscr$T(S). We show that this map extends to the Thurston compactification of $ \mathscr$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 $ \mathbb {CP}$1 structures on S. For each X $ \in$ $ \mathscr$T(S), we show that the boundary of the space P(X) of $ \mathbb {CP}$1 structures on X in the compactification of the grafting coordinates is the graph $ \Gamma$(iX) of the antipodal involution iX : $ \mathbb {P}$$ \mathscr$ML(S)$ \to$$ \mathbb {P}$$ \mathscr$ML(S).

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