A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation, which can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be considered as the Schwarzian derivative of a 1 structure, to which one can naturally associate another measured geodesic lamination using grafting.
We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomorphism ML (S)Q(X) for each conformal structure X on a compact surface S. We show that these maps are nearly the same, differing by a multiplicative factor of -2 and an error term of lower order than the maps themselves (which we bound explicitly).
As an application we show that the Schwarzian derivative of a 1
structure with Fuchsian holonomy is close to a 2-integral
Jenkins-Strebel differential. We also study compactifications of the
space of 1 structures using the Schwarzian derivative and
grafting coordinates; we show that the natural map between these
extends to the boundary of each fiber over Teichmüller space, and we
describe this extension.