Groups whose subgroups of infinite rank are closed in the profinite topology

Abstract

A subgroup X of a group G is closed in the profinite topology if it can be obtained as intersection of a collection of subgroups of finite index of G. It is proved that if all subgroups of infinite rank of a group G are closed, then either G has finite rank or all its subgroups are closed, provided that either G is nilpotent-by-finite or it has finite conjugacy classes.