 Nielsen–Thurston classification

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 Nielsen–Thurston classification is a way of classifying the types of homeomorphisms on the surface of a two-dimensional manifold, particularly in the context of surfaces with hyperbolic geometry. It specifically deals with the study of homeomorphisms of compact surfaces, particularly orientable and non-orientable surfaces.