Background:
I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ Here $\mathbb{D}$ denotes the open unit disk and $\mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right)$ is the plane minus the Gaussian integer lattice. I know this is possible, as $\mathbb{D}$ is the universal cover for the codomain above.
In his answer to this MathOverflow question, Prof. Eremenko described one method of doing this. However, I have little knowledge of conformal and hyperbolic geometry, so I am having trouble following his response. Intuitively, the argument is fairly clear, but I would like a more rigorous undertanding. I have summarised the procedure he gives (so far as I understand) below.
Prof. Eremenko's method:
- Choose the right ideal hyperbolic quadrilateral.
Consider the "circular quadrilateral" inscribed in the unit disk with vertices at $1, -1, i$ and $-i$. We will denote this quadrilateral by $Q$. The sides of $Q$ will be arcs of circles that are orthogonal to the unit disk. Another way to think of $Q$ is as the bounded region enclosed by an astroid (pictured below) inscribed in the unit disk. (Note As per Prof. Eremenko below, the astroid is not a valid way to think of this)
Considering $\mathbb{D}$ as the Poincaré disk, this is an ideal hyperbolic quadrilateral.
- Conformally map this to $\mathbb{C}$.
There is a conformal map $\varphi$ of this circular quadrilateral to a square with vertices at $0, 1, i$ and $1 + i$; we can choose this map to take vertices to vertices. Because the vertices of $Q$ are at infinity, the points $0, 1, i$ and $1 + i$ are not in the image $\varphi(Q)$.
- Apply Schwarz's reflection principle to extend $\varphi$ to a map $\mathbb{D} \to \mathbb{C}$.
Once this is done we have our continuous map from $\mathbb{D}$ to $\mathbb{C}$ with image set $\mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right)$.
My questions:
I am not entirely sure of the conformal map needed to take $Q$ to the square with vertices $0, 1, i, 1+i$. I believe I just need to choose the correct Möbius transformation, is this so? Or do I need another tool, like perhaps the Riemann mapping theorem.
I do not fully understand how Schwarz's reflection principle will allow me to extend this map to the entire unit disk, much less to a map with the desired properties (ie excluding the integer lattice).
Any help will be much appreciated!
Edited: to strikeout astroid remark.
First let me correct a mistake in your question: is it WRONG to think of this as an astroid. It is just 4 arcs of circles, NOT an astroid, though it looks somewhat similar.
Second. The conformal map is certainly NOT fractional-linear (which you call Mobius). This conformal map is written explicitly in the paper I already referred to on MO: arXiv:1110.2696. Of course, it exists by the Riemann mapping theorem, but the explicit expression also can be given, in terms of the hypergeometric function.
On your second question, the space allowed here is too small to lecture on the Schwarz reflection Principle, and I recommend that you just consult some book on Complex variable. The most comprehensive treatment of the relevant maps in is Caratheodory's textbook. Ahlfors book is also OK and there are many others.