It is well-known that $SL_2(\mathbb{Z})$ is generated by $S = \left( \begin{array}{ccc}0 & -1 \\1 & 0 \end{array} \right), T = \left( \begin{array}{ccc}1 & 1 \\0 & 1 \end{array} \right)$(e.g. Serre's A Course in Arithmetic). Let $\sigma \in SL_2(\mathbb{Z})$. Is there algorithm to explicitly write $\sigma$ as a product of $S, T^n, n \in \mathbb{Z}$? If yes, how?
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1Isn't the algorithm basically just the Euclidean algorithm? – Alex Youcis Dec 01 '13 at 03:29
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@AlexYoucis I have no idea how the Euclidean algorithm can be used to solve the problem. – Makoto Kato Dec 01 '13 at 03:33
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4I'm not sure if you can noodle around with stuff to only need to use one $S$, but look here: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,Z).pdf – Alex Youcis Dec 01 '13 at 03:34