20.85 (2022)

Let $F$ be a free group. An element $\omega \in F$ is said to be primitive if there is a minimal generating system of $F$ that contains $\omega$, almost primitive if it is primitive in each finitely generated proper subgroup of $F$ containing $\omega$, tame almost primitive if, whenever $\omega^\alpha$ is contained in a subgroup $H$ of $F$ with $\alpha \geqslant 1$ minimal, either $\omega^\alpha$ is primitive in $H$ or the index of $H$ in $F$ is just $\alpha$. In (B. Fine, A. Moldenhauer, G. Rosenberger, L. Wienke, Topics in Infinite Group Theory: Nielsen Methods, Covering Spaces, and Hyperbolic Groups, De Gruyter, Berlin, 2021) it is shown that $u = [a_1, b_1][a_2, b_2] \dots [a_g, b_g]$ is tame almost primitive in the free group on $a_1, b_1, \dots, a_g, b_g$ with $g \geqslant 1$, and $v = c_1^2 \dots c_p^2$ is tame almost primitive in the free group on $c_1, \dots, c_p$ with $p \geqslant 2$.

Are there tame almost primitive elements in free groups other than $u, v$, and their product $uv$ in the free group on $a_1, b_1, \dots, a_g, b_g, c_1, \dots, c_p$?