Assuming any facts about triangulations that you need, prove the Riemann-Hurwitz theorem.

Use the Riemann-Hurwitz theorem to prove that, for any cubic polynomial $f: \mathbb{C} \rightarrow \mathbb{C}$, there are affine transformations $g(z)=a z+b$ and $h(z)=c z+d$ such that $k(z)=g \circ f \circ h(z)$ is of one of the following two forms:

$$k(z)=z^{3} \text { or } k(z)=z\left(z^{2} / 3-1\right)$$