Weighted completion of Galois groups and Galois actions on the fundamental group

Fix a prime number l. In this paper we prove a conjecture [16, p. 300], which Ihara attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fundamental group of the thrice punctured projective line. It is stated below. Similar techniques are also used to prove part of a conjecture of Goncharov [11, Conj. 2.1], also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line, and which derives from the conjectures of Deligne and Ihara and questions of Drinfeld [7, p. 859]. Ihara’s version of Deligne’s conjecture concerns the outer action φl : GQ → Outπ1(P (C)− {0, 1,∞}, x) (1)