On the Galois Actions on the Fundamental Group of $\mathbf{P}^1_{\mathbf{Q}(\mu_n)} \backslash \{0,\mu_n,\infty\}$

Abstract

We are studying the action of Galois groups on the pro-$l$ completion of the fundamental group of $\mathbf{P}^1_{\overline{\mathbf{Q}(\mu_n)}} \backslash \{0,\mu_n,\infty\}$. If $n=2p$, where $p$ is an odd prime number then the Lie algebra of derivations associated to the image of $\text{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}(\mu_{2p\cdot l^\infty}))$ has $\frac{p-1}{2}$ generators in each even degree and $\frac{p-1}{2}$ generators in each odd degree greater than $1.$ We shall show that generators in even degrees generate a free Lie algebra.