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 $\mathbb {CP}$ 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) $\to$ 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 $\mathbb {CP}$ ¹ structure with Fuchsian holonomy is close to a 2 $\pi$ -integral Jenkins-Strebel differential. We also study compactifications of the space of $\mathbb {CP}$ 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.


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