Primitivity rank for random elements in free groups

TL;DR

This paper investigates the primitivity rank of elements in free groups, showing that most elements have maximal primitivity rank and are non-primitive only in the whole group, revealing typical structural properties.

Contribution

It establishes that a generic set of elements in free groups have maximal primitivity rank and are non-primitive only in the entire group, extending understanding of subgroup structures.

Findings

Most elements have primitivity rank equal to the group's rank

For generic elements, the only containing subgroup where they are non-primitive is the whole group

The set of such elements is exponentially generic

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 $Crit(w)=\{F_r\}$.