The Extrinsic Primitive Torsion Problem

TL;DR

This paper investigates the finiteness of quotient groups formed by primitive elements in free groups, fully characterizing certain cases and providing explicit representations for specific quotients.

Contribution

It establishes the conditions under which the quotient groups are finite and provides explicit characterizations and representations for key cases.

Findings

$F_2/P_k$ is finite only for $k=1,2,3$

Complete characterization of $F_2/P_k$ for $k=2,3,4$

Construction of a faithful nine-dimensional representation of $F_2/P_4$ with infinite image

Abstract

Let $P_k$ be the subgroup generated by $k$th powers of primitive elements in $F_r$, the free group of rank $r$. We show that $F_2/P_k$ is finite if and only if $k$ is $1$, $2$, or $3$. We also fully characterize $F_2/P_k$ for $k = 2,3,4$. In particular, we give a faithful nine dimensional representation of $F_2/P_4$ with infinite image.