ON ι-ADIC ITERATED INTEGRALS, IV -Ramification and generators of Galois actions on fundamental groups and torsors of paths

We are studying Galois representations on fundamental groups and on torsors of paths of a projective line minus a finite number of points. We reprove by explicit calculations some known results about ramification properties of such representations. We calculate the number of generators in degree 1 of the images of these Galois representations. We show also that the number of linearly independent generators in degree greater than 1 is equal &franc12 φ(n) for the action of G_Q(μ5) on the fundamental group of P¹_¯Q \ ({0,∞} ∪ μ_n). Finally we show that the graded Lie algebra associated with the action of G_Q(μ5) on the fundamental group of P¹_¯Q \ ({0,∞} ∪ μ₅) is not free.