to my result. The appendix was born out of an insightful discussion with Saul Schleimer on a bus ride and it would not have been written up without Mladen Bestvina’s encouragement. Last but not least, I thank my advisor Matt Clay for his constant support.

Definitions and Preliminaries {#defs}
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In this paper, free groups $F$ are assumed to have finite rank at least $2$. We will study the ascending HNN extension of a free group and how its properties relate to those of the defining endomorphism. Let $A \le F$ be a subgroup of a free group and $\phi : A \to F$ be an injective homomorphism, then we define the [**HNN extension**]{} of $F$ over $A$ to be: $$F*_A = \left\langle\, F, t~|~t^{-1} a t = \phi(a), \forall a \in A \,\right\rangle$$ An HNN extension has a natural map $F*_A \to \mathbb Z$ that maps $F \mapsto 0$ and $ t \mapsto 1$; we shall refer to this map as the [**natural fibration**]{}. For the rest of this paper, we restrict ourselves to HNN extensions defined over free factors. When A = F , then we call $F*_F = F*_\phi$ an [**ascending HNN extension**]{} or a [**mapping torus**]{} 