Riemann surfaces and the geometrization of 3-manifolds

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists, so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3-manifolds, which can be built up in an inductive way from 3-balls, i.e., {\em Haken} manifolds. Thurston's construction of a hyperbolic structure is also inductive. At the inductive step one must find the right geometry on an open 3-manifold so that its ends may be glued together. Using quasiconformal deformations, the gluing problem can be formulated as a fixed-point problem for a map of Teichm\"uller space to itself. Thurston proposes to find the fixed point by {\em iterating} this map. Here we outline Thurston's construction and sketch a new proof that the iteration converges. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently large covering space (e.g., the universal cover). This contraction gives an immediate estimate for the contraction of Thurston's iteration.