### 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 $4$.

Zdzislaw Wojtkowiak <zdzislaw@mpim-bonn.mpg.de>