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 pth 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.
Francesco Noseda; Ilir Snopce. On hereditarily self-similar $p$-adic analytic pro-$p$ groups. Groups, geometry, and dynamics, Tome 16 (2022) no. 1, pp. 85-114. doi: 10.4171/ggd/641
@article{10_4171_ggd_641,
author = {Francesco Noseda and Ilir Snopce},
title = {On hereditarily self-similar $p$-adic analytic pro-$p$ groups},
journal = {Groups, geometry, and dynamics},
pages = {85--114},
year = {2022},
volume = {16},
number = {1},
doi = {10.4171/ggd/641},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/641/}
}
TY - JOUR
AU - Francesco Noseda
AU - Ilir Snopce
TI - On hereditarily self-similar $p$-adic analytic pro-$p$ groups
JO - Groups, geometry, and dynamics
PY - 2022
SP - 85
EP - 114
VL - 16
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/641/
DO - 10.4171/ggd/641
ID - 10_4171_ggd_641
ER -
%0 Journal Article
%A Francesco Noseda
%A Ilir Snopce
%T On hereditarily self-similar $p$-adic analytic pro-$p$ groups
%J Groups, geometry, and dynamics
%D 2022
%P 85-114
%V 16
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/641/
%R 10.4171/ggd/641
%F 10_4171_ggd_641
