For a free group $F$, an element $w$ is primitive if it is part of some free basis for $F$.


Let $\pi:F[x\_0,x\_1,...,x\_n]\rightarrow F[x\_1,x\_2,...,x\_n]$ be defined $\pi (x\_0)=1$ and $\pi (x\_i)=x\_i$ for $i\geq 1$.


My question is:


Is there an example of a cyclically reduced primitive element $w$ in $F[x\_0,x\_1,...,x\_n]$ satisfying $\pi(w)=1$ whose length is greater than 1?


[Added for clarification]


Note: if $w$ is an example, then there must be more than one occurrence of $x\_0$ in $w$ and the exponent sum of $x\_0$ in $w$ must be $\pm 1$.



Primitive elements in a free group with trivial projection
No, there cannot be a primitive element $w \in \ker \pi$ that is not conjugate to $x\_0$ or $x\_0^{-1}$.


The map $\pi$ factors through $F[x\_0, x\_1, \dots, x\_n] / \langle \langle w \rangle \rangle$, which is isomorphic to $F\_n$ since $w$ is primitive. As the free group is Hopfian, this means the induced surjection $F[x\_0, x\_1, \dots, x\_n] / \langle \langle w \rangle \rangle \to F\_n$ is in fact an isomorphism, so $w$ and $x\_0$ have the same normal closure in $F\_{n+1}$. A theorem of Magnus from 1931, citing from Lyndon and Schupp's book "Combinatorial Group Theory", states:


**Proposition II.5.8.** If two elements $r\_1$ and $r\_2$ in a free group $F$ have the same normal closure in $F$, then $r\_1$ is conjugate to $r\_2$ or to $r\_2^{-1}$.


