Primitivity rank for random elements in free groups

Autor:	Kapovich, Ilya
Rok vydání:	2021
Předmět:	Mathematics - Group Theory Mathematics - Geometric Topology
Druh dokumentu:	Working Paper
Popis:	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 $Crit(w)=\{F_r\}$. Comment: Updated with a reference to a paper of Cashen and Hoffmann and a discussion of their results
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2109.09400 Zobrazit plný text záznamu View this record from Arxiv