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Let $z$ be a point in a fundamental domain of $\Gamma(2)\subset \mathrm{SL}_2(\mathbf{Z})$ in the complex upper half plane.

Does there exist an $\epsilon >0$ such that the geodesic distance $d(z,\gamma z)$ satisfies

$$ \cosh( d(z,\gamma z)) > 1+\epsilon$$ for all $\gamma \neq \pm 1$ in $\Gamma(2)$?

Note that $\epsilon$ will probably depend on $z$.

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Assuming I understand correctly, the answer is yes. $\Gamma(2)$ is a discrete group, and acts properly discontinuously on the upper half-plane. Moreover, it acts freely, so no non-trivial $\gamma$ fixes any $z$. Thus you can find an $\epsilon$ sufficiently small that the ball of center $z$ and radius $\epsilon$ contains no other points in the orbit of $z$. (This is perhaps easier to see if you consider the Dirichlet fundamental domain centered at $z$, or at least if $z$ is in the interior of your fundamental domain).

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