It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all the integral lattice points) is the upper half plane $\mathcal{H}$.

How should I think about this map? How does the map behave near the missing lattice points?

A related question is this: $\mathcal{H}$ is also the universal cover for a punctured torus, whose fundamental group is $F_2$, the free group on two generators. By comparing the punctured torus to the wedge of two circles, I feel like the universal cover for the punctured torus, ie $\mathcal{H}$, ought to be deformation-retractable to an infinite 4-regular tree. Ie, the infinite 4-regular tree ought to be able to be embedded in $\mathcal{H}$ such that the vertices of the tree all lie on the boundary of $\mathcal{H}$. What does this tree look like in $\mathcal{H}$?

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