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

Abstract

Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with $\dim (G)<p$, 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 $p⩾5$, 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 $S{L}_1^1({\Delta }_p)$ admits such an action. On the other hand, we prove that all the open subgroups of $S{L}_2^{△}({\mathbb{Z}}_p)$ admit faithful self-similar actions on regular rooted trees.

Corrections to the statements of Conjecture 2.9, Conjecture 2.10, Lemma 2.11, and Lemma 2.12, and to the proof of Theorem D

Cited by in corpus (1)

• On pro-$p$ groups with quadratic cohomology