## 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.

### MSC:

 11C08 Polynomials in number theory 11R32 Galois theory
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