The case where S is a torus is handled separately and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic, and the tools of Teichmüller theory become useful. In what follows, we assume S has genus at least two, as this is the case Thurston considered. The three types in this classification are notmutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved union of simple closed curves Γ. Each of the resulting compact surfaceswith boundary is acted upon by some power of g, and the classification can again be applied to this homeomorphism.
Thurston's classification applies to homeomorphisms of orientable surfaces of genus ≥ 2, but the type of a homeomorphism only depends on its associated element of the mapping class groupMod. In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good geometric properties. For example:
When g is pseudo-Anosov, there is an element of its mapping class that preserves a pair of transverse singular foliations of S, stretching the leaves of one while contracting the leaves of the other.
Thurston's original motivation for developing this classification was to find geometric structures on mapping tori of the type predicted by the Geometrization conjecture. The mapping torusMg of a homeomorphism g of a surface S is the 3-manifold obtained from S × by gluing S × to S × using g. The geometric structure of Mg is related to the type of g in the classification as follows:
If g is periodic, then Mg has an H2 × R structure;
If g is reducible, then Mg has incompressible tori, and should be cut along these tori to yield pieces that each have geometric structures ;
If g is pseudo-Anosov, then Mg has a hyperbolic structure.
The first two cases are comparatively easy, while the existence of a hyperbolic structure on the mapping torus of a pseudo-Anosov homeomorphism is a deep and difficult theorem. The hyperbolic 3-manifolds that arise in this way are called fibered because they are surface bundles over the circle, and these manifolds are treated separately in the proof of Thurston's geometrization theorem for Haken manifolds. Fibered hyperbolic 3-manifolds have a number of interesting and pathological properties; for example, Cannon and Thurston showed that the surface subgroup of the arising Kleinian group has limit set which is a sphere-filling curve.
The three types of surface homeomorphisms are also related to the dynamics of the mapping class group Mod on the Teichmüller spaceT. Thurston introduced a compactification of T that is homeomorphic to a closed ball, and to which the action of Mod extends naturally. The type of an element g of the mapping class group in the Thurston classification is related to its fixed points when acting on the compactification of T:
If g is periodic, then there is a fixed point within T; this point corresponds to a hyperbolic structure on S whose isometry group contains an element isotopic to g;
If g is pseudo-Anosov, then g has no fixed points in T but has a pair of fixed points on the Thurston boundary; these fixed points correspond to the stable and unstable foliations of S preserved by g.
For some reducible mapping classes g, there is a single fixed point on the Thurston boundary; an example is a multi-twist along a pants decompositionΓ. In this case the fixed point of g on the Thurston boundary corresponds to Γ.
This is reminiscent of the classification of hyperbolic isometries into elliptic, parabolic, and hyperbolic types.
