Intersection of maximal subgroups in prosolvable groups

Abstract

Let $H$ be an open subgroup of a profinite group that can be expressed as intersection of maximal subgroups of $G.$ Given a positive real number $\eta ,$ we say that $H$ is an $\eta$-intersection if there exists a family of maximal subgroups $M_1,\dots ,M_t$ such that $H=M_1\cap \dots \cap M_t$ and $|G:M_1|\cdots |G:M_t|\le |G:H{|}^{\eta }$. We investigate the meaning of this property and its influence on the group structure.