论文标题
关于遗传自我相似的$ P $ -ADIC Analitaltic Pro- $ P $组
On hereditarily self-similar $p$-adic analytic pro-$p$ groups
论文作者
论文摘要
如果每个非平凡的有限生成的$ g $ $ g $的封闭亚组承认在$ p $ -ary的树上，$ g $ a $ p $ $ p $ $ g $ $ p $都非常相似，则据说是遗传性相似的，$ g $是强烈的，$ g $ a。我们对可解决的无扭转$ p $ -Adic Analytic Pro- $ P $组的尺寸小于$ p $，这些尺寸在索引$ p $上非常相似。此外，我们表明，可解决的无扭转$ p $ - 亚种分析pro- $ p $小于$ p $的尺寸小于$ p $，在索引$ p $的情况下非常相似，并且只有当它与最大$ p $ p $ p $ p $ p $ galois组同构时，其中包含一个原始的$ p $ p $ - p $ - p $ - p $ - the limity lotity的词。为了证明上述结果的关键步骤，我们对3维解决的无扭转$ P $ -ADIC分析Pro- P $组进行了分类，该组承认对$ P $ -ary树采取了忠实的自我相似的动作，完成了3-维无托管$ P $ P $ P $ P $ -Adic Analytic Pro-P $组的分类。
A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of dimension less than $p$ that are strongly hereditarily self-similar of index $p$. Moreover, we show that a solvable torsion-free $p$-adic analytic pro-$p$ group of dimension less than $p$ is strongly hereditarily self-similar of index $p$ if and only if it is isomorphic to the maximal pro-$p$ Galois group of some field that contains a primitive $p$-th root of unity. As a key step for the proof of the above results, we classify the 3-dimensional solvable torsion-free $p$-adic analytic pro-$p$ groups that admit a faithful self-similar action on a $p$-ary tree, completing the classification of the 3-dimensional torsion-free $p$-adic analytic pro-$p$ groups that admit such actions.
