The dihedral Lie algebras and Galois symmetries of \pi_1^l(P^1 - 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. 
A.B. Goncharov