Profinite iterated monodromy groups arising from quadratic polynomials
Profinite iterated monodromy groups arising from quadratic polynomials
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由二次多项式产生的有限迭代单数群
DOI：
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发表时间：
2013
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通讯作者：
R. Pink
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作者：
R. Pink
我们详细研究了作为任意二次多项式在不同于两个的特征域上的几何'etale迭代单调群而产生的有限群G。这是正则有根二叉树自同构群的自相似闭子群。 （当基域为 C 时，它是通常拓扑结构的有限生成迭代单向群的闭包，这也经常被研究。）除此之外，我们证明了 G 的共轭类和同构类仅取决于多项式的后临界轨道的组合类型。
我们通过显式递归定义的生成器来表示 G 的选定实例。共轭性的唯一性取决于某种半刚性属性，这确保了这些生成元在树的自同构群下的任意共轭总是生成与 G 共轭的子群。我们使用进一步的显式生成元来确定 Hausdorff 维数、最大阿贝尔因子群和 G 的归一化器。然后使用归一化器的描述来描述二次多项式的算术'etale迭代单数群。
使用的方法纯粹是群论，根本不涉及 C 上的基本群。
We study in detail the profinite group G arising as geometric 'etale iterated monodromy group of an arbitrary quadratic polynomial 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. (When the base field is C it is the closure of the finitely generated iterated monodromy group for the usual topology which is also often studied.) Among other things we prove that the conjugacy class and hence the isomorphism class of G depends only on the combinatorial type of the post-critical orbit of the polynomial.
We represent a chosen instance of G by explicit recursively defined generators. The uniqueness up to conjugacy depends on a certain semirigidity property, which ensures that arbitrary conjugates of these generators under the automorphism group of the tree always generate a subgroup that is conjugate to G. We determine the Hausdorff dimension, the maximal abelian factor group, and the normalizer of G using further explicit generators. The description of the normalizer is then used to describe the arithmetic 'etale iterated monodromy group of the quadratic polynomial.
The methods used are purely group theoretical and do not involve fundamental groups over C at all.