Fundamental group of the torus using the edge group

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I'm following the notes of Lackenby to study Algebraic Topology and he gives the prove of the fundamental group of the circle using the edge group. Since I already know the triangulation of the torus I was trying to prove that its fundamental group is $\mathbb{Z} \times\mathbb{Z}$ but I don't know how to start. Can anyone give me a hint, please?