# Nielsen–Thurston classification

In mathematics, **Thurston's classification theorem** characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen (1944).

Given a homeomorphism *f* : *S* → *S*, there is a map *g* isotopic to *f* such that at least one of the following holds:

The case where *S* is a torus (i.e., a surface whose genus is one) is handled separately (see torus bundle) 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. (Note, however, that the cases where *S* has boundary or is not orientable are definitely still of interest.)

The three types in this classification are **not** mutually 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 surfaces *with boundary* is acted upon by some power (i.e. iterated composition) 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 group *Mod(S)*. In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good geometric properties. For example:

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 (also due to Thurston). 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(*S*) on the Teichmüller space *T*(*S*). Thurston introduced a compactification of *T*(*S*) that is homeomorphic to a closed ball, and to which the action of Mod(*S*) 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*(*S*):

This is reminiscent of the classification of hyperbolic isometries into *elliptic*, *parabolic*, and *hyperbolic* types (which have fixed point structures similar to the *periodic*, *reducible*, and *pseudo-Anosov* types listed above).