Why $\Bbb{C}/G \cong \Bbb{C}^*/G^*$?

Question

Let $G = \{g \mid g: z\mapsto z +k+l\omega ,\text{ with } k,l \in \Bbb{Z}\}$ (where $\text{Im}\ \omega >0$) be group of translation acting on $\Bbb{C}$.

Under the exponential map $\exp(2\pi i):\Bbb{C}\to \Bbb{C}^*$ we have another action $G^* = \{g^* \mid g^*:z\mapsto z\cdot e^{2\pi i n\omega} , \text{with } n \in \Bbb{Z}\}$ which is action on $\Bbb{C}^*$, and we can check that expoential map given above is compatible with the $G$ and $G^*$ action that is $$\exp(2\pi i)(g\cdot z) = g^*\cdot \exp(2\pi i)(z)$$

with $g(z) = z+ m_1\omega + m_2 \in G$ and $g^* = e^{2\pi i m_1 \omega} \in G^*$

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I want to prove that quotient space are the same that is :

$$\Bbb{C}/G \cong \Bbb{C}^*/G^*$$

I know if $f:X\to Y$ is $G$ - equivariant homeomorphism then $X/G \cong Y/G$ but here the expoential map is no longer homeomorphism, how to gurantee that it's still induced homeomorphism on the quotient?

Answer 1

$\newcommand{\Numbers}[1]{\mathbf{#1}}\newcommand{\Cpx}{\Numbers{C}}\newcommand{\Ints}{\Numbers{Z}}$As in the question, fix a complex number $\omega$ with positive imaginary part (and, if desired without loss of generality, satisfying $|\operatorname{im}\omega| \leq \frac{1}{2}$ and $1 \leq |\omega|$). Introduce the additive subgroup $$ G = \Ints + \Ints\omega = \{k + \ell\omega : k,\ \ell \in \Ints\} \subset (\Cpx, +) $$ acting by translations, and the multiplicative subgroup $$ G^{*} = \{\exp(2\pi in\omega) : n \in \Ints\} \subset (\Cpx^{\times}, \cdot) $$ acting by complex multiplication.

Details to fill in as needed:

• $G^{*} = \exp(G)$. Specifically, $\exp$ induces a surjection from $G = \Ints + \Ints\omega$ to $G^{*}$ with kernel $\Ints$.
• The covering map $\exp:\Cpx \to \Cpx^{\times}$ followed by the quotient map $\Cpx^{\times} \to \Cpx^{\times}/G^{*}$ factors through the quotient, inducing a bijection $\Cpx/G \to \Cpx^{\times}/G^{*}$. Because the group actions are via biholomorphisms and $\exp$ is a holomorphic covering map (hence a local biholomorphism), this bijection is a biholomorphism.
• Regarding the without loss of generality claim, every integer lattice in $(\Cpx, +)$ is, up to scaling and rotation, generated by $1$ and a complex number $\omega$, the modulus, satisfying $|\operatorname{im}\omega| \leq \frac{1}{2}$ and $1 \leq |\omega|$, and two such lattices are equal if and only if their moduli differ by $\pm1$ or are negative reciprocals. (Geometrically, the boundary of the modular domain is glued up by folding along the center line.)

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Bart de Smit and Hendrik Lenstra used these ideas in a 2004 article in the AMS Notices to analyze the Droste effect in M.C. Escher's print Print Gallery. There used to be mpeg animations hosted at Leiden of infinite-zooming into the center of Escher's print based upon "unwrapping"—lifting Escher's lithograph to the universal cover—then mapping back to $\Cpx^{\times}$ after scaling and rotation. The only thing that turns up in a quick web search, however, is this youtube video https://youtu.be/9WHdyG9mJaI