Index-stable compact p-adic analytic groups · arXivDesk
2006.08000Jun 14, 2020Main body by F. Noseda and I. Snopce. Added: Appendix by J-P. Serre; Remark after Theorem 1; Theorem 9; Remark 10; Corollary 11. Modified: 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 Qp-Lie algebra L(G)
Nearby in the stack
. We prove that
G
is index-stable whenever
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