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Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}_p$. I am looking for an answer or existing literature on the following question:

What is known about the structure of the maximal pro-$\ell$-quotient of the absolute Galois group of $K$? Is it finitely generated? Torsion-free?

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  • $\begingroup$ Is $\ell \neq p$ assumed? $\endgroup$ – Matt Papanikolas Jun 5 '18 at 14:47
  • $\begingroup$ Yes, assume $\ell \neq p$. $\endgroup$ – darko Jun 6 '18 at 7:53
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I assume your field has characteristic $p>0$. Then the maximal pro-$l$ quotient of the absolute Galois group is torsion-free. Indeed, by results of E. Becker, Euklidische Korper und euklidische Hullen von Korpern, J. reine angew. Math. 278-269 (1974), 41-52. He shows that torsion elements in such quotients can only be involutions arising from orderings on the ground field. But in positive characteristic there are no orderings.

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