Grafting Rays Fellow Travel Teichmüller Geodesics

Given a measured geodesic lamination on a hyperbolic surface, grafting the surface along multiples of the lamination defines a path in Teichmüller space, called the grafting ray. We show that every grafting ray, after reparameterization, is a Teichmüller quasi-geodesic and stays in a bounded neighborhood of some Teichmüller geodesic. As part of our approach, we show that grafting rays have controlled dependence on the starting point; that is, for any measured geodesic lamination λ, the map of Teichmüller space which is defined by grafting along λ is L-Lipschitz with respect to the Teichmüller metric, where L is a universal constant. This Lipschitz property follows from an extension of grafting to an open neighborhood of Teichmüller space in the space of quasi-Fuchsian groups.