Free-by-cyclic groups are equationally Noetherian

Abstract 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 et al. (Ann Math 161:1–59, 2005) [6] are 2-acylindrical, when the growth is at least quadratic.