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Arboreal Galois representations. (English) Zbl 1206.11069
Summary: Let $$G_{\mathbb{Q}}$$ be the absolute Galois group of $$\mathbb{Q}$$, and let $$T$$ be the complete rooted $$d$$-ary tree, where $$d \geq 2$$. 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 $$\operatorname{Aut}(T)$$ can occur as the image of an arboreal representation of $$G_{\mathbb{Q}}$$.

##### MSC:
 11F80 Galois representations 11R32 Galois theory 20E08 Groups acting on trees 20E18 Limits, profinite groups
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##### References:
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