First examples of non-abelian quotients of the Grothendieck-Teichmueller group that receive surjective homomorphisms from the absolute Galois group of rational numbers

Many challenging questions about the Grothendieck-Teichmueller group, $GT$, are motivated by the fact that this group receives the injective homomorphism (called the Ihara embedding) from the absolute Galois group, GQ, of rational numbers. Although the question about the surjectivity of the Ihara embedding is a very challenging problem, in this paper, we construct a family of finite non-abelian quotients of $GT$ that receive surjective homomorphisms from GQ. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from GQ to the resulting profinite group is also surjective. We give an explicit description of this profinite group. To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the gentle version of the Grothendieck-Teichmueller group. This groupoid was introduced in the recent paper by the second author and J. Guynee and the set $Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$ of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral poset. We show that each element K of $Dih$ is the only object of its connected component in $GTSh$. Using the surjectivity of the cyclotomic character, we prove that, if the order of the dihedral group corresponding to K is a power of 2, then the natural homomorphism from GQ to the finite group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target K in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that the natural homomorphism from GQ to the finite group $GTSh(K, K)$ is surjective for every object K of the dihedral poset.