  We study compact hyperbolic surface laminations. These are a generalization
of closed hyperbolic surfaces which appear to be more suited to the study of
Teichm\"uller theory than arbitrary non-compact surfaces. We show that the
Teichm\"uller space of any non-trivial hyperbolic surface lamination is
infinite dimensional. In order to prove this result, we study the theory of
deformations of hyperbolic surfaces, and we derive what we believe to be a new
formula for the derivative of the length of a simple closed geodesic with
respect to the action of grafting. This formula complements those derived by
McMullen in [23], in terms of the Weil-Petersson metric, and by Wolpert in
[33], for the case of earthquakes.
