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.
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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 , , and for its action on first homology.