Fundamental group of $\mathbb{C^*}/G$

Question

Calculate the fundamental group of $\mathbb{C^*/G}$ where $G=\{\phi^n:n\in \mathbb{Z}\}$ is the group of the homeomorphisms s.t. $\phi (z)=2z$ and $\mathbb{C^*}=\mathbb{C}-\{0\}$

Any hints on how to attack this exercise? thanks

Answer 1

Hint Construct a homeomorphism $\Phi : \Bbb C^* \stackrel{\cong}{\to} \Bbb S^1 \times \Bbb R_+$ through which $G$ factors into a (continuous) action on the second factor, or more precisely, for which there is an action $\ast$ of $G$ on $\Bbb R_+$ such that $$\Phi(g \cdot z) = (\Phi_1(z), g \ast \Phi_2(z)) $$ for all $g \in G$. Then, $$\pi_1(\Bbb C^* / G) \cong \pi_1(\Bbb S^1 \times (\Bbb R_+ / G)) \cong \pi_1(\Bbb S^1) \times \pi_1(\Bbb R_+ / G),$$ so it remains only to understand the simpler quotient space $\Bbb R_+ / G$.