The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ where $S_d$ is the permutation group on $d$ symbols. If this homomorphism is transitive, i.e. the image of $\phi$ acts transitively on $\{1, \ldots ,d\}$, then this data allows one to construct a unique connected Riemann surface $Y$ with a map $f: Y \to X$ which is a branched cover: when we restrict $f$ to $f^{-1}(X \backslash \Delta)$ it is a $d$-fold cover and around the branch points the monodromy is given by $\phi$. This is known as the Riemann existence theorem and it is proven by constructing the cover corresponding to kernel of $\phi$ and filling in the missing points with disks using $\phi$. If $X$ is compact, $Y$ will be as well.
My question is: given the Riemann sphere (or any other Riemann surface), which compact connected Riemann surfaces can one get if one is allowed to pick $\Delta$, $d$ and $\phi$ as above? This may seem like a trivial question: topologically any Riemann surface arises this way. But it is not clear to me what complex structures can arise. Alternatively, the question may be phrased as: is the map from such data to the disjoint union of moduli spaces of Riemann spaces of different genus surjective?
I would of course also be interested in literature discussing this or related questions.
Edit: the answers are correct that this question was easy. I was actually interested in the situation where we demand that $\phi$ assigns to a loop around a point in $\Delta$ permutation of consisting of disjoint cycles of length two, such that we get only branching of degree 2.
So I guess the updated question is: what restrictions can we place on $\phi$ such that we still get all compact connected Riemann surfaces this way?
Best Answer
The answer is simple: any complex structure arises in this way.
Indeed, any compact Riemann surface $X$ admits a holomorphic cover $\phi \colon X \to \mathbb{P}^1$.
This is straightforward in genus $0$ and $1$. If the genus is at least $2$, then the linear system $|3K_X|$ is very ample, so it gives an embedding $\gamma \colon X \to \mathbb{P}^N$ and the map $\phi$ is obtained by composing $\gamma$ with a suitable projection.
We are using here the fact that every compact Riemann surface is a smooth complex projective curve; this follows from the existence of a meromorphic function on it, which is a non-trivial fact (see Andy Putman's comment).
EDIT. We can always chose $\phi$ so that the local monodromy is general, that is it is given by a single transposition around each branch point. In fact, projecting $X \subset \mathbb{P}^N$ from a general subspace of the right codimension, we obtain a plane curve $X' \subset \mathbb{P}^2$ whose singularities are at worst ordinary double points. Moreover, $X'$ has at most a finite number of bitangent lines; taking the projection $\pi_p \colon X' \to \mathbb{P}^1$ from a point $p$ not contained in any of the bitangent lines and composing with the map $X \to X'$ we obtain a finite cover $\phi \colon X \to \mathbb{P}^1$ whith the desired property.