Let $\Sigma\_g$ be a surface of genus $g\ge 2$, and let $\Sigma\_k$ be an $m$-sheeted covering 
space of $\Sigma\_g$. It is known that $k=m(g-1)+1$. 


An example of such a covering space is a regular covering obtained by choosing one ``hole" as the center of the symmetry and take $\Sigma\_k$ to have $m$-fold rotational symmetry around that chosen center (as on standard pictures in your favorite topology book).


**Question:** Does every finite sheeted regular covering space of $\Sigma\_g$ arise in this way? 


It feels like this should be known/standard, but I can't find an argument or a reference. 


**Edit**: I agree, the question is not very precise as stated, I certainly didn't have in mind that every finite cover arises via cyclic symmetry as in the example above. The reason for the lack of precision is that I do not have a particular result in mind, but I would like to know if there is a simple geometric description of the covering *map*, as Misha points out, between two surfaces? The example with the cyclic group gives such a simple description of the covering map.
Or, given any covering map $\Sigma\_h\to \Sigma\_g$ between two surfaces, is there some kind of a ``standard" covering $M\to \Sigma\_g$, which factors through $\Sigma\_h$?



Covering spaces of surfaces

> 
> Given any covering map $\Sigma\_h\to\Sigma\_g$ between two surfaces, is there some kind of a ``standard" covering $M\to \Sigma\_g$, which factors through $\Sigma\_h$? 
> 
> 
> 


In brief, the answer to this part of the question is 'You can take $M\to\Sigma\_g$ to be regular, but beyond that, no.' This can already be extracted from Misha's comments above, but let me try to summarise.


First, note that there is a regular covering $M\to\Sigma\_g$ that factors through $\Sigma\_h\to\Sigma\_g$. (Specifically, you can take $\pi\_1M$ to be the intersection of all the conjugates of $\pi\_1\Sigma\_h$.) So, as in the earlier part of your question, you can take $\Sigma\_h\to\Sigma\_g$ to be regular.


You can now rephrase your question in terms of normal quotients of $\pi\_1\Sigma\_g$, and it becomes



> 
> Given any finite quotient $q:\pi\_1\Sigma\_g\to Q$, is there some kind of 'standard' finite quotient $p:\pi\_1\Sigma\_g\to P$ such that $q$ factors through $p$?
> 
> 
> 


In particular, $P$ surjects $Q$. But *any* finite group can arise as $Q$ (see Misha's and algori's comments---the point is that $\pi\_1\Sigma\_g$ surjects a free group), so you are looking for a 'standard' family of finite groups that surjects every finite group. But there's no 'natural' definition of such a family. 


**Remark:** Obviously, there are such families, such as $\{ Q\times\mathbb{Z}/2\}$ where $Q$ is an arbitrary finite group, but clearly this is not 'natural'.


