Iterated Grafting and Holonomy Lifts of Teichmueller space

Abstract

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