The dihedral Lie algebras and Galois symmetries of π₁^l(P¹ - 0, infinity and N-th roots of unity)

We 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.