In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand we show that several infinite families of subgroups of the automorphism group of the infinite binary tree cannot appear as images of arboreal representations of quadratic polynomials over number fields, yielding unconditional evidence towards Jones' finite index conjecture. On the other hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-critically finite. Finally, we combine our results with local class field theory to prove that if $f\in \mathbb Q[x]$ is quadratic and $\alpha\in \mathbb Q$, then the dynamical Galois group attached to the pair $(f,\alpha)$ is abelian if and only if $(f,\alpha)$ is $\mathbb Q$-conjugated to either $(x^2,\pm1)$ or $(x^2-2,\beta)$, where $\beta\in \{\pm 2,\pm 1,0\}$. This improves on recent results of Andrews and Petsche.

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二次树栖表示的约束图像

在本文中，我们证明了附加到二次多项式的有限生成动态伽罗瓦群的几个结果。首先，我们表明，在全局域上，二次后临界有限多项式正是那些具有树栖表示的，其图像是拓扑有限生成的。为了得到这个结果，我们还证明了 Hindes 的动力学非等平凡性猜想的二次情况。接下来，我们给出这个结果的两个应用。一方面，我们证明无限二叉树的自同构群的几个无限子群族不能作为二次多项式在数域上的树形表示的图像出现，从而为琼斯的有限指数猜想提供了无条件的证据。另一方面，我们证明了具有阿贝尔动力学伽罗瓦群的全局域上的二次多项式必然是后临界有限的。最后，我们将我们的结果与局部类场论相结合，证明如果 $f\in \mathbb Q[x]$ 是二次的且 $\alpha\in \mathbb Q$，那么附加到对 $( f,\alpha)$ 是阿贝尔当且仅当 $(f,\alpha)$ 是 $\mathbb Q$-共轭到 $(x^2,\pm1)$ 或 $(x^2-2,\ beta)$，其中 $\beta\in \{\pm 2,\pm 1,0\}$。这改进了 Andrews 和 Petsche 最近的结果。\alpha)$ 是 $\mathbb Q$ 与 $(x^2,\pm1)$ 或 $(x^2-2,\beta)$ 共轭，其中 $\beta\in \{\pm 2, \pm 1,0\}$。这改进了 Andrews 和 Petsche 最近的结果。\alpha)$ 是 $\mathbb Q$ 与 $(x^2,\pm1)$ 或 $(x^2-2,\beta)$ 共轭，其中 $\beta\in \{\pm 2, \pm 1,0\}$。这改进了 Andrews 和 Petsche 最近的结果。