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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 \(G_{\mathbb{Q}}\).

11F80 Galois representations
11R32 Galois theory
20E08 Groups acting on trees
20E18 Limits, profinite groups
Full Text: DOI
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