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I'm trying to think of two matrices $A,B\in SL_2(\Bbb{Z})$ of finite order ($A^n=B^m=I$) with the property that $AB=C$ where the order of $C$ is infinite.

I guess that just by trial and error I could find two of those matrices, but I would like to find those in a little bit more sophisticated way. What would be a good strategy ?

Kasper
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    Its one of the answers here ---> http://math.stackexchange.com/questions/314850/example-of-a-group-where-oa-and-ob-are-finite-but-oab-is-infinite/314857#comment681871_314857 – Sean Ballentine Feb 27 '13 at 17:16

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Besides to what @Clayton pointed at above link you can take the following matrices as well: $$A=\left(\begin{array}[cc] .0 & -1\\ 1 & 0\end{array}\right)\quad\text{and}\quad B=\left(\begin{array}[cc].0&1\\ -1 & -1\end{array}\right)$$ You can check that $A^4=I=B^3$ but $AB$ has infinite order. This means that $tSL_2(\mathbb Z)$ including of all torsion elements of the group is not a subgroup.

Mikasa
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