Consider a family $ \lbrace T_i \rbrace_{i\in\mathbb{Z}} $ of tori indexed on the integers such that every torus is attached to the preceding one and to the subsequent by a single point. How can we compute the fundamental group of this infinite row of tori?
I want to stress the fact that «every torus is attached to the preceding one and to the subsequent by a single point» therefore, this is not a duplicate of the question Fundamental Group is free on infinite generators.
The fundamental group is the free product of the groups $\pi_1(T_i)$ for $i \in \mathbb{Z}$, and so it is isomorphic to a free product of countably infinite copies of the group $\mathbb{Z} \oplus \mathbb{Z}$.
To show this, the first step is to show that the fundamental group of the union of the tori $T_i$ for $-n \le i \le n$ is the free product of $\pi_1(T_i)$ for $-n \le i \le n$, and this is done by induction using Van Kampen's theorem. The second step is done just using a direct limit argument.