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

### Francesco Noseda

Federal University of Rio de Janeiro, Brazil### Ilir Snopce

Federal University of Rio de Janeiro, Brazil

## Abstract

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.

## Cite this article

Francesco Noseda, Ilir Snopce, On hereditarily self-similar $p$-adic analytic pro-$p$ groups. Groups Geom. Dyn. 16 (2022), no. 1, pp. 85–114

DOI 10.4171/GGD/641