Accessing non-abelian quotients of the Grothendieck-Teichmueller group via elementary tools
Ivan Bortnovskyi, Vasily A. Dolgushev, Borys Holikov, Vadym Pashkovskyi
TL;DR
This work connects the absolute Galois group $G_{\mathbb{Q}}$ to non-abelian quotients of the profinite Grothendieck–Teichmüller group $\widehat{\mathsf{GT}}$ by exploiting GT‑shadows in the groupoid $\mathsf{GTSh}$. It introduces the dihedral poset $\mathsf{Dih}$ via kernels of explicit maps $\psi_n$ from $\mathrm{PB}_3$ to dihedral groups and proves that each associated finite shadow group $\mathsf{GT}(\mathsf{K}^{(n)})$ is a non-abelian quotient of $G_{\mathbb{Q}}$ for suitable $n$, notably when $n=2^{\alpha}$ with $\alpha\ge2$, where all shadows are arithmetical and the Ihara map is surjective. The results yield an infinite non-abelian profinite quotient $\lim\mathcal{ML}|_{\mathsf{Dih}_2}$ of both $G_{\mathbb{Q}}$ and $\widehat{\mathsf{GT}}$, explicitly realized as a clopen subgroup of $\mathrm{Aff}(\mathbb{Z}_2)$. The Lochak–Schneps conditions provide practical tests for shadow genuineness, and the paper conjectures surjectivity of $G_{\mathbb{Q}}$ onto $\mathsf{GT}(\mathsf{K})$ for every object in the dihedral poset. Overall, the work furnishes concrete, elementary tools to access non-abelian quotients of $\widehat{\mathsf{GT}}$ and highlights a pathway to broader arithmetical GT phenomena.
Abstract
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, $G_Q$, 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 $G_Q$. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$ 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 $G_Q$ 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 $G_Q$ to the finite group $GTSh(K, K)$ is surjective for every object $K$ of the dihedral poset.