On the self-similarity of the norm one group of $p$-adic division algebras

Abstract

Let $p$ be a prime, $K$ a finite extension of $\mathbb{Q}_p$, $D$ a finite dimensional central division $K$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When $p\geqslant \mathrm{dim}(SL_1(D))$, we provide an infinite family of congruence subgroups of $SL_1(D)$ that do not admit self-similar actions on regular rooted $p$-ary trees. The proof requires results on $\mathbb{Z}_p$-Lie lattices that also lead to the classification of the torsion-free $p$-adic analytic pro-$p$ groups $G$ of dimension less than $p$ with the property that all the nontrivial closed subgroups of $G$ admit a self-similar action on a $p$-ary tree. As a consequence we obtain that a nontrivial torsion-free $p$-adic analytic pro-$p$ group $G$ of dimension less than $p$ is isomorphic to the maximal pro-$p$ Galois group of a field that contains a primitive $p$-th root of unity if and only if all the nontrivial closed subgroups of $G$ admit a self-similar action on a regular rooted $p$-ary tree.Comment: 25 page