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Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.

How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $N \rightarrow \infty$?

Any suggestions?Please also provide either a reference or a short argument.

2 added 9 characters in body

Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.

How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $\Gamma$ grow?N \rightarrow \infty\$?

Any suggestions? Please also provide either a reference or a short argument.

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