We know that a complex polynomial $P$ of degree $n$ is an $n$-sheeted covering from $$\{\mathbb{C} - P^{-1}\{\text{critical values of }P\}\} \to \{\mathbb{C} - \{\text{critical values of }P\}\}. $$ So the question is how to determine whether such a cover is regular or not, and its group of deck transformations.
The specific example I was trying was $P(Z)=Z^3 - 3Z$. The critical values are $\{2,-2\}$. So the group is $F_2$, i.e. the free group on 2 generators, and $$P^{-1}\{\text{critical values of }P\} = \underset{\text{corr. to }2}{\{2,-1\}} \cup \underset{\text{corr. to }-2}{\{-2,1\}}.$$ So here the group is $F_4$. I see the facts that a small simple loop around $2$ maps to a simple loop around $2$, and small simple loop around $-1$ maps to a double loop around $2$. And similar for preimage of $-2$. We know that if the image of the fundamental group of domain is normal in fundamental group of codomain, then the covering is regular. So can we fix a point and say that a simple loop just going around 2 will map to a simple loop going around 2 and similarly for other points? I guess not (because then the image would be whole group, which contradicts the fact that it is a $3$-sheeted cover).
So how to proceed? Can anybody help?
Since one has a $3$-sheeted (unbranched holomorphic) covering of a connected Riemann surface, the deck group $G$ has at most $3$ elements (see Remark to Definition 5.5, Forster - Lectures on Riemann Surfaces).
Let $\varphi$ be a (non trivial) deck transformation, by definition: \begin{equation} \forall z\in\mathbb{C}\setminus\{\pm1,\pm2\}\equiv Y,\,\varphi(z)^3-3\varphi(z)=z^3-3z\\ \varphi(z)^3-z^3=3[\varphi(z)-z]\\ [\varphi(z)-z][\varphi(z)^2+z\varphi(z)+z^2]=3[\varphi(z)-z]\\ \varphi(z)^2+z\varphi(z)+z^2-3=0\\ \varphi(z)=\frac{-z\pm|12-3z^2|e^{\textstyle\frac{arg(12-3z^2)}{2}i}}{2}; \end{equation} by all this: \begin{equation} G=\left\{Id_Y,\varphi_{-}(z)=\frac{-z-|12-3z^2|e^{\textstyle\frac{arg(12-3z^2)}{2}i}}{2},\varphi_{+}(z)=\frac{-z+|12-3z^2|e^{\textstyle\frac{arg(12-3z^2)}{2}i}}{2}\right\}\cong\mathbb{Z}_3, \end{equation} where $|\cdot|$ and $arg(\cdot)$ are the module and the mainvalue of $\cdot$, respectively.