Primitive elements in free groups

Abstract:

Let ${F_n}$ denote the free group of rank $n$ generated by ${x_1},{x_2}, \ldots ,{x_n}$. We say that $y \in {F_n}$ is a primitive element of ${F_n}$ if it is contained in a set of free generators of ${F_n}$. In this note we construct, for each integer $n \geq 4$, an $(n - 1)$-generator group $H$ that has an $n$-generator, $2$-relator presentation $H = \langle {x_1}, \ldots ,{x_n}|{r_1},{r_2}\rangle$ such that the normal closure of $\{ {r_1},{r_2}\}$ in ${F_n}$ does not contain a primitive element of ${F_n}$.