A group G is said to be equationally Noetherian if every system of equations over G has the same solution set as some finite subsystem. This property, introduced in the 1990s in the context of algebraic geometry over groups, has found its way into logic over groups and geometric group theory, in particular the study of limit groups and acylindrically hyperbolic groups. In this talk, I will explain why all extensions G of a finitely generated free group by the infinite cyclic group are equationally Noetherian. As a consequence, the set of exponential growth rates for any such G is well-ordered. The proofs are based on various group actions on trees.
This is joint work with Monika Kudlinska.
