Free-by-cyclic groups are equationally Noetherian

A group $G$ is said to be equationally Noetherian if every system of equations in $G$ is equivalent to a finite subsystem. We show that all free-by-cyclic groups are equationally Noetherian. As a corollary, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered. Along the way, we prove that free-by-cyclic groups with polynomially growing monodromies of infinite order admit non-elementary 4-acylindrical actions on trees. We show that the splittings arising from the improved relative train track machinery of Bestvina-Feighn-Handel are 2-acylindrical when the growth is at least quadratic.