On hereditarily self-similar $p$-adic analytic pro-$p$ groups

Abstract

A non-trivial finitely generated pro-p group G is said to be strongly hereditarily selfsimilar 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.

On self-similarity of p-adic analytic pro-p groups of small dimension

“…For the rest of this paragraph, assume that p 5. In [Nos19] we classified the 3-dimensional unsolvable torsion-free p-adic analytic pro-p groups, and we determined which of them are self-similar of index p. On the other hand, in [Nos20] we determined which 3-dimensional solvable torsion-free p-adic analytic pro-p groups are self-similar of index p. Moreover, we established that every 3-dimensional solvable torsion-free p-adic analytic pro-p group is self-similar of index p 2 ; as a consequence, the self-similarity index of any such group is less or equal to p 2 . It is worth mentioning that the results of [Nos19] do not exclude the possibility that the self-similarity index of self-similar 3-dimensional unsolvable torsion-free p-adic analytic pro-p groups is bounded; indeed, they do not even exclude the possibility that this bound is p 2 .…”