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THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 22 / Issue 2 / March 2023
- Published online by Cambridge University Press:
- 29 September 2021, pp. 961-983
- Print publication:
- March 2023
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- Article
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For a prime number p and a free profinite group S on the basis X, let
$S_{\left (n,p\right )}$, 
$n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For 
$p>n$, we give a word-combinatorial description of the cohomology group 
$H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.
