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

## TL;DR

This paper classifies small-dimensional torsion-free $p$-adic analytic pro-$p$ groups based on their self-similar actions on rooted trees, linking Lie lattice structures with group actions, and identifies specific groups with such properties.

## Contribution

It provides a classification of 3-dimensional unsolvable $p$-adic analytic pro-$p$ groups and their self-similar actions using Lie lattice endomorphisms, extending Lazard's correspondence.

## Key findings

- Classified 3-dimensional unsolvable $bZ_p$-Lie lattices for odd p.
- Determined which groups admit faithful self-similar actions on p-ary trees.
- Showed that some groups like $SL_1^1( riangle_p)$ do not admit such actions.

## Abstract

Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with $\mathrm{dim}(G) < p$, we show that the self-similar actions of $G$ on regular rooted trees can be studied through the virtual endomorphisms of the associated $\mathbb{Z}_p$-Lie lattice. We explicitly classify 3-dimensional unsolvable $\mathbb{Z}_p$-Lie lattices for $p$ odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free $p$-adic analytic pro-$p$ groups for $p\geqslant 5$, and to determine which of them admit a faithful self-similar action on a $p$-ary tree. In particular, we show that no open subgroup of $SL_1^1(\Delta_p)$ admits such an action. On the other hand, we prove that all the open subgroups of $SL_2^{\triangle}(\mathbb{Z}_p)$ admit faithful self-similar actions on regular rooted trees.