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Let $H$ be the upper half plane. For a finite index subgroup $\Gamma\le PSL(2,\mathbb{Z})$, the closed geodesics on the modular curve $H/\Gamma$ correspond to conjugacy classes of cyclic subgroups generated by hyperbolic elements (i.e., |trace| $\ge 3$). Here, by a geodesic I mean the image of a geodesic on $H$ in $H/\Gamma$.

I'd like to understand how to detect when a given hyperbolic element determines a simple geodesic on $H/\Gamma$.

As I understand this is a rather difficult problem. What are some examples of subgroups $\Gamma$ for which this question has a nice (and nontrivial) answer?

An example of a trivial answer would be that whenever $H/\Gamma$ is a thrice punctured sphere, there are no simple closed geodesics.

For example, is there a good answer for $\Gamma = PSL(2,\mathbb{Z})$?

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  • $\begingroup$ This is indeed a difficult problem in general. One very classical case is when $\Gamma$ is a once-punctured torus. In this particular case, it is a theorem of Nielsen (I think) that the simple closed curves are exactly the elements of the fundamental group that are elements of a minimal generating set. These are exactly the Christoffel words, and there is a nice recipe for them (which I learned from Marco Linton). I'm not sure about $PSL_2(\mathbb{Z})$, though. $\endgroup$ Commented Feb 13 at 16:41

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