I admit this is a *very* broad question, but I am looking for general properties of **[finitely generated free]-by-[infinite cyclic]** groups. More precisely, what are some properties that the groups $\{F_n\rtimes_\phi\mathbb{Z}\ |\ n\geq 2,\ \phi\in Aut(F_n)\}$ have in common?

These are some properties that *not* all of those groups have (even though some do):

- They are not all hyperbolic.
- They are not all $\mathrm{CAT}(0)$ groups.

These are some properties that the groups *do* have in common:

- They all have solvable conjugacy problem.
- They all satisfy a quadratic isoperimetric inequality.

Furthermore for $n=2$, all groups of the form $F_2\rtimes_\phi\mathbb{Z}$ are strongly poly-free. This will not be true in higher rank however, as for $n\geq 3$ not all automorphisms of $F_n$ are geometric.

I understand that the structure of $F_n\rtimes_\phi\mathbb{Z}$ highly depends on $\phi$, so fully understanding all common properties probably boils down to fully understanding $Aut(F_n)$.