Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits
Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits
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由具有无限后临界轨道的二次态射产生的有限迭代单峰群
DOI：
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发表时间：
2013
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通讯作者：
R. Pink
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文献类型：
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作者：
R. Pink
本文详细研究了特征不等于2的域上具有无穷后临界轨道的任意二次态射f的几何迭代单值群G.这是一个自相似闭子群的群的自同构的正规根二叉树。在许多情况下，它等于树的自同构群，但仍有一些有趣的情况下，它不是。在这些情况下，我们证明了G的共轭类只依赖于f的后临界轨道的组合类型。我们还确定了G的Hausdorff维数和正规化子。这个结果被用来描述f的算术迭代单值群。
该方法主要使用组理论和相同类型的前一篇文章中的同一作者处理二次多项式与有限的后临界轨道。抽象自相似profinite群作用在正则根二叉树上的结果可能是独立的兴趣。
We study in detail the profinite group G arising as geometric \'etale iterated monodromy group of an arbitrary quadratic morphism f with an infinite postcritical orbit over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. In many cases it is equal to the automorphism group of the tree, but there remain some interesting cases where it is not. In these cases we prove that the conjugacy class of G depends only on the combinatorial type of the postcritical orbit of f. We also determine the Hausdorff dimension and the normalizer of G. This result is then used to describe the arithmetic \'etale iterated monodromy group of f.
The methods used mostly group theoretical and of the same type as in a previous article of the same author dealing with quadratic polynomials with a finite postcritical orbit. The results on abstract self-similar profinite groups acting on a regular rooted binary tree may be of independent interest.