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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 $\pi_1(\Sigma_2)$?)

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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 $g=2$.

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 $\Pi_g\to G$.

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  • $\begingroup$ That will do it. In particular, every simple group is a quotient of the fundamental group of a genus-two surface. $\endgroup$
    – Dmitry Vaintrob
    Commented Aug 1, 2019 at 3:53
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    $\begingroup$ By the way, your map $\pi_1\Sigma_g\to F_g$ can be understood geometrically, by embedding $\Sigma_g$ as the boundary of a solid three-dimensional handlebody (which retracts to a bouquet of circles). $\endgroup$
    – Dmitry Vaintrob
    Commented Aug 1, 2019 at 5:34

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