Continuity of Thurston's length function

Abstract.

A measured lamination μ geodesically realized in a hyperbolic 3-manifold M has a well-defined average length, due to W. Thurston. For $M \cong S \times {\Bbb R}$ we prove that the function measuring the average length of the maximal realizable sublamination of μ varies bicontinuously in M and μ. Since connected, positive, non-realizable measured laminations arise as zeros of the length function, its continuity suggests new behavioral features of quasi-isometry invariants under limits of hyperbolic 3-manifolds.