Free-by-cyclic groups are equationally Noetherian
Abstract
A group is said to be equationally Noetherian if every system of equations in 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.
Cite
@article{arxiv.2407.08809,
title = {Free-by-cyclic groups are equationally Noetherian},
author = {Monika Kudlinska and Motiejus Valiunas},
journal= {arXiv preprint arXiv:2407.08809},
year = {2025}
}
Comments
16 pages, comments welcome! v2: expanded the introduction and abstract; v3: minor changes based on the referee's comments; final version, to appear in Mathematische Annalen