Iterated Grafting and Holonomy Lifts of Teichmueller space

Abstract

Let $X$ be a closed hyperbolic surface and $\lambda, \eta$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $\lambda$ and $\eta$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $\lambda$ and $\eta$. Using this result, we study the holonomy lifts $gr_{\lambda}\rho_{X,\lambda}$ of Teichmueller geodesics $\rho_{X,\lambda}$ for integral laminations $\lambda$ and show that all of them have bounded Teichmueller distance to the geodesic $\rho_{X,\lambda}$. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences $\gr_{n\lambda}X$ and show that they converge geometrically to a punctured surface.