Let $\Sigma\_g$ be the a closed orientable surface of genus $g$. 


My somewhat naive question: what is known about simple finite factors of $\pi\_1(\Sigma\_g)?$ In particular, I know that the composition series of this group is well understood in terms of homological algebra and topology: does this series filter $\pi\_1(\Sigma\_g)$? (In which case, all simple factors would be abelian.) Or conversely, is every finite simple group a quotient of some $\pi\_1(\Sigma\_g)?$ (Or even a quotient of $\pi\_1(\Sigma\_2)$?)



What finite simple groups appear as factors of surface fundamental groups?

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> It appears that you are asking which groups $G$ occur as quotients of the fundamental group $\Pi\_g=\pi\_1(\Sigma\_g)$ for some $g$. Then the answer is "all finitely generated $G$", where $g$ can be taken to be the rank of $G$, i.e. the least cardinality of a generating set. If $G$ is a finite simple group, then it is known to be generated by 2 elements, so we can take $g=2$.
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*Proof* Using the standard one-relator presentation of 
$\Pi\_g$ as 
$$\langle a\_1,\ldots,a\_g,b\_1,\ldots,b\_g\,|\,[a\_1,b\_1]\cdots [a\_g,b\_g]\rangle,$$
the map sending $a\_i, 1\leq i\leq g\,$ to some generators of $G$ and each $b\_i$ to the identity of $G$ uniquely extends to a group surjection $\Pi\_g\to G$. 


