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 $\mathcal{H}$?
Best Answer
On the first question (the universal cover of the complement of a lattice). The missing points are in the image, so it is not the map that "behaves" but the inverse map. The inverse map behaves in a very simple way: it has infinitely many "logarithmic singularities" over each missing point. "How to think about the map" is not a well defined question. But the way I think about it is this. Consider the circular quadrilateral inscribed in the unit disc, say with vertices 1,-1,i,-i; the sides are arcs of circles orthogonal to the unit circle. All angles of this quadrilateral are 0. There is a conformal homeomorphism of this circular quadrilateral onto a (rectilinear) square with vertices 0,1,1,1+i, sending vertices to vertices. By Schwarz's symmetry principle, applied very many times, the map extends to a map from the unit disc to the plane minus the lattice. This is our universal covering map. You can make a picture. You can express it in terms of special functions (it is a ratio of two solutions of a very special Heun equation, linear differential equation of second order with regular singular points at 1,-1,i,-1.
EDIT: I am not sure what exactly you want to know, in your question you mention visualization, rather than computation, but once I computed this map, arXiv:1110.2696. It can be expressed in terms of hypergeometric functions. And I also asked a MO question related to it: Maximum of a function of one variable.