Arboreal Galois representations

Abstract

Let $G_{\mathbb{Q}}$ be the absolute Galois group of $\mathbb{Q}$, and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of $G_{\mathbb{Q}}$ into the automorphism group of T, particularly in the case d =  2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of $G_\mathbb{Q}$. We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of $G_{\mathbb{Q}}$. Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of $G_{\mathbb{Q}}$.