Index-stable compact $${\varvec{p}}$$-adic analytic groups

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.