On the Galois actions of torsors of paths I, by Zdzislaw Wojtkowiak

We are studying representations obtain from actions of Galois groups on torsors of paths on a projective line minus a finite number of points. Using these actions on torsors of paths we construct geometrically representations of Galois groups which realize $\ell$-adically the associated graded Lie algebra of the fundamental group of the tannakien category of mixed Tate motives over ${\rm Spec}\, \Zbb$, ${\rm Spec}\, \Zbb [\frac{1}{q}]$, ${\rm Spec}\,\Oc _{\Qbb ( \sqrt {-q})}$ for any prime number $q$ ($q\neq 2$ in the last case) and over ${\rm Spec}\,\Oc _{\Qbb ( \sqrt {-q})}[\frac{1}{q}]$ for any prime number $q$ congruent to $3$ modulo $4$.

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