Intersections of maximal subgroups in prosolvable groups

Given a positive real number we say that an open subgroup H of a profinite group G is an -intersection if there exists a family of maximal subgroups such that and . We study the finitely generated prosolvable groups G with the property that there exists such that every maximal intersection in G is an -intersection. We prove in particular that this is equivalent to a condition on the centralizers in G of the complemented chief factors of G. We use the notion of -intersection to address an open question asking whether the number of maximal intersections of index n in a finitely generated prosolvable group G is polynomially bounded. We prove that the answer is positive if the derived subgroup of G is pronilpotent.