TITLE: Relative/acylindrical hyperbolicity of free-by-cyclic groups
QUESTION [7 upvotes]: Is this statement true?

Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.
If $\varphi$ is polynomially growing and of infinite order, then the semidirect product $\mathbb{F}\rtimes_\Phi\mathbb{Z}$
is acylindrically hyperbolic
but not relatively hyperbolic.

If there is anything unclear about notation, please tell me. This statement is closely related to Problem 8.2 in A. Minasyan and D. Osin. Acylindrical hyperbolicity of groups acting on trees.
math. Annalen, 362:1055–1105, 2015 (arXiv link). This problem asks which mapping tori of injective endomorphisms of finitely generated free
groups are acylindrically hyperbolic.

REPLY [5 votes]: In addition to Henry's answer, I would like to mention that the acylindrical hyperbolicity of free-by-cyclic groups is fully characterised in the recent preprint Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups, allowing us to remove the word virtually in Button and Kropholler's statement. Actually, we have the more general proposition:
Theorem: Fix a group $H$ and a morphism $\varphi : H \to \mathrm{Aut}(\mathbb{F}_n)$ where $n \geq 2$. The semidirect product $\mathbb{F}_n \rtimes_\varphi H$ is acylindrically hyperbolic if and only if 
$$\mathrm{ker} \left( H \overset{\varphi}{\to} \mathrm{Aut}(G) \to \mathrm{Out}(G) \right)$$
is a finite subgroup of $H$.
In the particular case where $H= \mathbb{Z}$, we find that $\mathbb{F}_n \rtimes_\Phi \mathbb{Z}$ is acylindrically hyperbolic if and only if $\Phi$ has infinite order in $\mathrm{Out}(\mathbb{F}_n)$. 
Such a behaviour seems to be quite common. The same statements hold for infinitely-ended groups, non-elementary hyperbolic groups, most non-elementary relatively hyperbolic groups (including toral relatively hyperbolic groups), irreducible right-angled Artin/Coxeter groups (and more generally, most graph products of groups; see this preprint).