论文标题
限制二次树木代表的图像
Constraining images of quadratic arboreal representations
论文作者
论文摘要
在本文中，我们证明了有限生成的二次多项式的有限生成的动态galois基团。首先，我们表明，在全球字段上，二次有限的有限多项式恰恰是那些具有固定图像的树木代表的人，其图像在拓扑上是有限生成的。为了获得此结果，我们还证明了Hindes对动态非遗传性的猜想的二次情况。接下来，我们给出两个结果的应用。一方面，我们证明，与Abelian Dynalial Galois组的全球领域上的二次多项式一定是批判性有限的，我们将结果与本地阶级田间理论相结合，以对$ \ MATHBB Q $进行分类，以超过$ \ Mathbb Q $与Abelian Dynalialical Galois组，以改进Andrews和Petsche的最新成绩。另一方面，我们表明，无限二进制树的自动形态群的几个无限族不能以数字领域的二次多项式的树木代表形式出现，从而为琼斯的有限索引猜想提供了无条件的证据。
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 prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-critically finite, and we combine our results with local class field theory to classify quadratic pairs over $\mathbb Q$ with abelian dynamical Galois group, improving on recent results of Andrews and Petsche. On the other 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.
