Free normal subgroup of an HNN-extension

by Amontillado   Last Updated September 19, 2019 21:20 PM - source

Suppose $F$ is a finitely generated free group and $a,b$ are not in $F'$ but $b^{-1}a \in F'$. By taking the HNN extension $G=\langle F,t | t^{-1}atb^{-1}\rangle$, is there a way to find a normal free subgroup of $G$ so that their quotient is cyclic?

I'm trying to define a homomorphism from $G$ to $\Bbb Z$ so that the kernel acts freely on the vertices of the HNN tree, which have the conjugates of $F$ as stabilizers but with no success. Moreover by defining $f_1:F \to \Bbb Z$ in general and let $f_2:\langle t\rangle \to \Bbb Z$ be trivial I have a map from $G\to \Bbb Z$ but this map can't be injective on the conjugates of $F$ because $f_1$ can never be injective due to $f_1(a)=f_2(b)$.

Is there something I'm missing?

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