Probabilistic properties of profinite groups

Abstract

Let $\mathfrak{C}$ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group $G$ and an element $x\in G$, we denote by $P_{\mathfrak{C}}(x,G)$ the probability that $x$ and a randomly chosen element of $G$ generate a pro-$\mathfrak{C}$ subgroup. We say that a profinite group $G$ is $\mathfrak{C}$-positive if $P_{\mathfrak{C}}(x,G)>0$ for all $x\in G.$ %Moreover we say that $G$ is $\mathfrak{C}$-bounded-positive if there exists a positive constant $\eta$ such that $P_{\mathfrak{C}}(x,G)>\eta$ for all $x\in G.$ We establish several equivalent conditions for a profinite group to be $\mathfrak{C}$-positive when $\mathfrak{C}$ is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite $\mathfrak{C}$-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.