Iterated grafting and holonomy lifts of Teichmüller space

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 Teichmüller 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 grλρX,λ of Teichmüller geodesics ρX,λ for integral laminations λ and show that all of them have bounded Teichmüller distance to the geodesic ρX,λ. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences grnλX and show that they converge geometrically to a punctured surface.