Index-stable compact p-adic analytic groups

摘要

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 Qp-Lie algebra L(G). We prove that G is index-stable whenever L(G) is semisimple. In particular, a justinfinite 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 L(G) has p-adic norm 1.