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.

中文翻译：

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在遗传自相似 $p$-adic 分析 pro-$p$ 组

如果 $G$ 的每个非平凡有限生成闭子群都承认在$p$-ary 树。我们对可解的无扭转 $p$-adic 分析 pro-p$ 组进行分类，其维度小于 $p$，它们在指数 $p$ 上具有很强的遗传自相似性。此外，我们证明了一个维数小于 $p$ 的可解无扭转 $p$-adic 解析 pro-p$ 群在指数 $p$ 上是强遗传自相似的，当且仅当它同构于最大值pro-$p$ 某个域的伽罗瓦组，它包含一个原始的 $p$th 统一根。作为证明上述结果的关键步骤，