I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}_n * \mathbb{Z}_2$ or even $\mathbb{Z}_m \rtimes \mathbb{Z}_n$.

This might be an interesting problem because I have written semidirect product $\rtimes$ rather than the free product $*$. A torus knot $K$ is defined in Hatcher as the image of an embedding of a map $f : S^1 \to S^1 \times S^1 \to \mathbb{R}^3 \subset S^3$ given by $z \mapsto (z^m, z^n)$ then the fundamental group $\pi_1(\mathbb{R}^3 - K)$ is $Z_m \ast Z_n$ possibly up to some number-theoretic conditions. Hatcher doesn't quite give you the answer.

I think the semidirect product $\mathbb{Z}_m \rtimes \mathbb{Z}_n$ is unique. We have to specify $\mathbb{Z}_m \lhd G$ and then $G = \mathbb{Z}_m \ltimes \mathbb{Z}_n$.

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