Defining R and G(R) - Segal, D; Tent, K (EMS Press, 2022)
We show that for Chevalley groups G(R)G(R) of rank at least 2 over an integral domain RR each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that RR and G(R)G(R) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields.