# Hatcher 1.3.25: finding $\pi_1$ of the quotient of the punctured place by a hyperbolic $\mathbb{Z}$-action

Problem 1.3.25 in Hatcher's "Algebraic Topology" is concerned with the quotient (call it $Y$) of $X=\mathbb{R}^2 \setminus \{0\}$ under the $\mathbb{Z}$-action generated by $(x,y) \mapsto (2x,y/2)$. Some parts of the problem are obvious to me ($Y$ is not Hausdorff, $Y$ is the union of four cylinders $\cong S^1 \times \mathbb{R}$) but I can't find $\pi_1(Y)$, although it fits into the exact sequence: $$1\to\mathbb{Z}\to\pi_1(Y)\to\mathbb{Z}\to1$$ from proposition 1.40(c) ($G\cong\pi_1(Z/G)/p_\ast (\pi_1(Z))$ for a cover $p: Z\to Z/G$ coming from a "covering space action" of $G$ on $Z$). Perhaps if I read the proofs more carefully I could unpackage this isomorphism, but I don't see how to find $\pi_1(Y)$ at the moment.

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