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:



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> 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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maximal pro-l-quotients of absolute Galois groups
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.


