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In the comments on this question, Robert Israel proved that the order of an element in $GL_2(\mathbb{Z})$ can be $2,3$ or $6$ (or infinite). This result is remarkedly reminiscent of the crystallographic restriction theorem, but I can't seem to find the relation. Is this mere coincidence, or is there something deep buried here?

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    They, of course, follow triviałly.2012-03-20

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The connection is that any lattice in $\mathbb{R}^2$ is isomorphic to $\mathbb{Z}^2$, so any (linear) group of symmetries of a lattice injects into $\text{GL}_2(\mathbb{Z})$.

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    Thanks. One has also to add two lines about why it is also valid for lattices in $\mathbb{R}^3$, but it's fairly easy.2012-03-20