Defining $R$ and $G(R)$

Abstract

We show that for Chevalley groups $G(R)$ of rank at least 2 over an integral domain $R$ each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that $R$ and $G(R)$ are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields.

For such groups it then follows that the group $G(R)$ is (finitely) axiomatizable in the appropriate class of groups provided $R$ is (finitely) axiomatizable in the corresponding class of rings.