The Extrinsic Primitive Torsion Problem

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.

34 pages, 5 figures, error corrections and exposition improvements following a thorough referee report

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