# 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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