Nielsen–Thurston classification theorem

The Nielsen–Thurston classification says that an orientation-preserving surface homeomorphism is periodic, reducible, or pseudo-Anosov. For a once-punctured torus, the three cases correspond to $|\mathrm{tr}M|<2$, $|\mathrm{tr}M|=2$, and $|\mathrm{tr}M|>2$ for its action $M\in S{L}_2(\mathbb{Z})$ on first homology.

The interior of the mapping torus of a pseudo-Anosov homeomorphism of a compact surface with boundary admits a complete finite-volume hyperbolic metric. Periodic monodromy instead gives a Seifert fibered mapping torus, while reducible monodromy gives a manifold containing an essential torus.