The dihedral Lie algebras and Galois symmetries of $\pi_1^{(l)}(\mathbb{P}-(\{0,\infty\}\cup \mu_N))$

Abstract

We describe the image of the absolute Galois group acting on the pro-$l$ completion of the fundamental group of the $\mathbb {G}_m$ minus $N$th roots of unity. We relate the structure of the image with geometry and topology of modular varieties for the congruence subgroups $\Gamma_1(m;N)$ of ${\rm GL}_m(\mathbb {Z})$ for $m=1,2,3\ldots$.