The image of an arboreal Galois representation. (English) Zbl 1167.11011

For a polynomial \(f\in \mathbb Q[X]\) denote by \(G(f)\) the Galois group of the field generated by all roots of all iterates of \(f\). If \(\deg f=d\), then the set of these roots may be identified with vertices of the \(d\)-ary rooted tree \(T\) and this leads to a continuous homomorphism \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \operatorname{Aut}(T)\) whose image is \(G(f)\). The authors determine \(G(f)\) for the generic quadratic polynomial \(f(x)=(x-t)^2+t+m\in \mathbb Q(t)[x]\) (\(m\in \mathbb Z\), \(m\neq-1\)), conjecture that the group \(G(f)\) for a specialization of \(f\) is usually a subgroup of a finite index in \(G(f)\) and prove this in some special cases. They formulate also other interesting conjectures.


11C08 Polynomials in number theory
11R32 Galois theory
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