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Dmitry Vaintrob
Dmitry Vaintrob
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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_g)$$\pi_1(\Sigma_2)$?)

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 $\pi_1(\Sigma_g)$?)

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 $\pi_1(\Sigma_2)$?)

standardized notation somewhat
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Dmitry Vaintrob
Dmitry Vaintrob
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Let $\Pi_g$$\Sigma_g$ be the fundamental group of a closed Riemannorientable surface of genus $g$.

My somewhat naive question: what is known about simple finite factors of $\Pi_g?$$\pi_1(\Sigma_g)?$ In particular, I know that the composition series of $\Pi_g$this group is well understood in terms of homological algebra and topology: does this series filter $\Pi_g$$\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_g?$$\pi_1(\Sigma_g)?$ (Or even a quotient of $\Pi_2$$\pi_1(\Sigma_g)$?)

Let $\Pi_g$ be the fundamental group of a closed Riemann surface of genus $g$.

My somewhat naive question: what is known about simple finite factors of $\Pi_g?$ In particular, I know that the composition series of $\Pi_g$ is well understood in terms of homological algebra and topology: does this series filter $\Pi_g$? (In which case, all simple factors would be abelian.) Or conversely, is every finite simple group a quotient of some $\Pi_g?$ (Or even a quotient of $\Pi_2$?)

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 $\pi_1(\Sigma_g)$?)

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

Let $\Pi_g$ be the fundamental group of a closed Riemann surface of genus $g$.

My somewhat naive question: what is known about simple finite factors of $\Pi_g?$ In particular, I know that the composition series of $\Pi_g$ is well understood in terms of homological algebra and topology: does this series filter $\Pi_g$? (In which case, all simple factors would be abelian.) Or conversely, is every finite simple group a quotient of some $\Pi_g?$ (Or even a quotient of $\Pi_2$?)