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THE NIELSENTHURSTON CLASSIFICATION 369 4. There is an affine representative Homeo + (T 2 ) of f with (F u , µ u ) = (F u , µ u ) and (F s , µ s ) = (F s , -1 µ s ), where > 1 is the leading eigenvalue of A. In this case we say that f is Anosov. The discussion so far can be summa- rized by the following. T HEOREM 13.1 Each nontrivial element f Mod(T 2 ) is of exactly one of the following types: periodic, reducible, Anosov. We can be even more specific in the first two cases. A nontrivial finite- order element of Mod(T 2 ) has order 2, 3, 4, or 6. Also, a nontrivial re- ducible element of Mod(T 2 ) is either a power of a Dehn twist or the product of a power of a Dehn twist with the hyperelliptic involution. The linear algebra approach. Using just the isomorphism Mod(T 2 ) SL(2, Z), and without appealing to hyperbolic geometry, we can give a more algebraic approach to the classification for Mod(T 2 ). Let A SL(2, Z) and let f Mod(T 2 ) denote the corresponding mapping class. The characteris- 2