Thurston ’ s work on surfaces

This book is an exposition of Thurston’s theory of surfaces: measured foliations, the compactification of Teichmuller space and the classification of diffeomorphisms. The mathematical content is roughly the following. For a surface M (let us say closed, orientable, of genus g > 1), one denotes by S the set of isotopy classes of simple closed curves in M . For α, β ∈ S, one denotes by i(α, β) the minimum number of geometric intersection points of α′ with β′, where α′ (resp. β′) is a simple curve in the class α (resp. β). This induces a map i∗ : S → R+ which turns out to be injective. In fact, if one projectivizes R+ \ 0, then i∗ induces an injection i∗ : S → P (R+) which endows S with a nontrivial topology. Here R+ is endowed with the weak topology ( = product topology). Two curves α, β ∈ S are “close” in P (R+) if, up to a multiple, they are made up of more or less the same strands going more or less the same direction. This is very different from saying that the curves are homotopic. The limits of curves are naturally interpreted as projective classes of “measured foliations”, that is, foliations that have an “invariant” transverse distance, and that have certain kinds of singularities (wellknown in the theory of quadratic differentials, or in smectic liquid crystals). The space of measured foliations considered in R+ (or in P (R+)) is denoted by MF (resp. PMF). One shows that MF ≃ R6g−6 and PMF ≃ S6g−7. In P (R+), the space PMF(M) and the Teichmuller space T (M) glue together into a (6g − 6)-dimensional ball: T (M) = T (M) ∪ PMF(M) = D6g−6. The group Diff(M) acts continuously on this compactification of T (this is hence a “natural” compactification). Hence any φ ∈ Diff(M) has a fixed point in T (M) (Brouwer) and the analysis of this fixed point shows that (up to isotopy) each φ is either a hyperbolic isometry of M , is “Anosov-like” (the word is “pseudo-Anosov”), or else is