Primitivity rank for random elements in free groups

Abstract: For a free group $F_r$ of finite rank $r\ge 2$ and a nontrivial element $w\in F_r$ the \emph{primitivity rank} $\pi(w)$ is the smallest rank of a subgroup $H\le F_r$ such that $w\in H$ and that $w$ is not primitive in $H$ (if no such $H$ exists, one puts $\pi(w)=\infty$). The set of all subgroups of $F_r$ of rank $\pi(w)$ containing $w$ as a non-primitive element is denoted $Crit(w)$. These notions were introduced by Puder in \cite{Pu14}. We prove that there exists an exponentially generic subset $V\subseteq F_r$ such that for every $w\in V$ we have $\pi(w)=r$ and $Crit(w)={F_r}$.