Finite index subgroups in profinite groups. (Q1409724)

The authors announce the following important result: in a finitely-generated profinite group all subgroups of finite index are open. This extends Serre's theorem for finitely-generated pro-\(p\) groups from the 1970s and is an extension which has long been sought for. It follows that for a finitely-generated profinite group the topology is uniquely determined by its group structure. The authors also prove that the terms of the (algebraic) lower central series of a finitely generated profinite group are closed. The proofs depend on properties of certain verbal subgroups and are outlined in this announcement.