Free-by-cyclic groups are equationally Noetherian
Monika Kudlinska, Motiejus Valiunas
TL;DR
This work proves that every free-by-cyclic group $G = F \rtimes_{\Phi} \mathbb{Z}$ is equationally Noetherian, extending algebraic-geometric methods over groups to a broad class of groups defined by monodromy dynamics. The authors develop a unified acylindrical-action framework on trees, analyze linear- and polynomial-growth monodromies via Bass–Serre splittings, and apply non-divergent ultrafilter arguments to establish equational Noetherianity; for exponential growth, relative hyperbolicity reductions and Groves–Hull techniques complete the proof. A key consequence is that the set of exponential growth rates $\xi(G)$ is well ordered for free-by-cyclic groups, aligning with recent results for hyperbolic and acylindrically hyperbolic groups. The results connect detailed automorphism dynamics (UPG, train tracks) with geometric group theory to yield structural and growth-rate insights for mapping tori of free-group automorphisms.
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-Feighn-Handel are 2-acylindrical when the growth is at least quadratic.