Index-stable compact $p$-adic analytic groups

TL;DR

This paper proves that certain compact p-adic analytic groups are index-stable based on the semisimplicity of their associated Lie algebras, providing a partial answer to a question by C. Reid.

Contribution

It establishes a criterion for index-stability of compact p-adic analytic groups linked to the semisimplicity of their Lie algebras, and characterizes just-infinite groups in this context.

Findings

Index-stability holds when the Lie algebra is semisimple.

A just-infinite compact p-adic analytic group is index-stable iff not virtually abelian.

Automorphisms have determinants with p-adic norm 1 if and only if the group is index-stable.

Abstract

A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let $p$ be a prime, and let $G$ be a compact $p$-adic analytic group with associated $\mathbb{Q}_p$-Lie algebra $\mathcal{L}(G)$. We prove that $G$ is index-stable whenever $\mathcal{L}(G)$ is semisimple. In particular, a just-infinite compact $p$-adic analytic group is index-stable if and only if it is not virtually abelian. Within the category of compact $p$-adic analytic groups, this gives a positive answer to a question of C. Reid. In the Appendix, J-P. Serre proves that $G$ is index-stable if and only if the determinant of any automorphism of $\mathcal{L}(G)$ has $p$-adic norm 1.