Index-stable compact $p$-adic analytic groups

Abstract

Let p be a prime, and let G be a compact p-adic analytic group with semisimple associated Q p -Lie algebra. We prove that any two isomorphic closed subgroups of G have the same index. In particular, a just-infinite compact p-adic analytic group contains a pair of open subgroups of different indices that are isomorphic to each other if and only if it is virtually abelian. Within the category of compact p-adic analytic groups, this gives a positive answer to a question of C. Reid.