Nielsen–Thurston classification

Remarks

The statement is about the elements of the mapping class group of $S$: each mapping class is periodic, reducible or pseudo-Anosov, and the third kind excludes the first two. A mapping class can be both periodic and reducible. The theorem also holds for compact surfaces with boundary and for surfaces with finitely many punctures.

Example: on the torus, mapping classes correspond to matrices $A$ in the special linear group $SL(2, \Z)$. If $\abs{\tr A} < 2$, then $A$ has finite order and the class is periodic. If $\abs{\tr A} = 2$, then $A$ has an integer eigenvector, and the class preserves the corresponding simple closed curve. For $A \neq \pm I$ it is a power of a Dehn twist up to sign and is reducible. If $\abs{\tr A} > 2$, then $A$ has real eigenvalues $\lambda^{\pm 1}$ with $\abs{\lambda} > 1$, and the class is Anosov, a pseudo-Anosov class without singularities.

Thurston's proof considers the action of $f$ on a compactification of the Teichmüller space of $S$ by projective measured foliations. This compactification is a closed ball, and the proof uses the Brouwer fixed-point theorem. A fixed point inside the ball gives a periodic class. A fixed point on the boundary gives either a reducing curve system or the stable and unstable foliations of a pseudo-Anosov map. Bers gave another proof by minimizing a quasiconformal dilatation over the isotopy class.

Going deeper

For a student

The theorem is the surface analogue of the classification of $2 \times 2$ integer matrices into elliptic, parabolic and hyperbolic ones. By cutting along the reducing curves and repeating, every mapping class decomposes into periodic and pseudo-Anosov pieces on subsurfaces. This decomposition controls the dynamics: the least topological entropy in an isotopy class is the largest $\log \lambda$ of its pseudo-Anosov pieces. By Thurston's hyperbolization theorem for fibered 3-manifolds, if $S$ has genus at least $2$, the mapping torus of $f$ has a hyperbolic structure exactly when $f$ is pseudo-Anosov.