2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59922We study the action of the Galois group on the pro-l-completion of the fundamental group of P^1 - {0, infinity and N-th roots of unity}. We describe the Lie algebra of the image of the Galois action and relate with the geometry of the modular varieties for GL_m for m = 1,2,3,... This story is the l-adic side of the motivic theory of multiple polylogarithms at roots of unity, which generalize the classical cyclotomy theory.80 pages, extended version of the paper appeared in 1998 in K-theory archiveAlgebraic GeometryNumber Theory11G65 11R32 11T22text