On galois automorphisms of the fundamental group of the projective line minus three points (Q914775)

Let \(U={\mathbb{P}}_ k^ 1-\{0,1,\infty\}\) and assume \(k\) is a number field of finite degree over \({\mathbb{Q}}\). For a fixed odd prime \(\ell\), let \(\pi'_ 1(U)\) denote the profinite fundamental group of \(U\) divided by the kernel of the geometric fundamental group to its pro-\(\ell\) completion \(\pi_ 1(U\otimes \bar k)^{pro-\ell}\). In this paper, it is shown that every group automorphism of \(\pi'_ 1(U)\) respecting the canonical augmentation to \(Gal(\bar k/k)\) must come from a \(k\)-automorphism of \(U\) up to inner automorphisms of \(\pi_ 1(U\otimes \bar k)^{pro-\ell}\).