On hereditarily self-similar \(p\)-adic analytic pro-\(p\) groups (Q2141703)

Summary: 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.