Regular bi-interpretability and finite axiomatizability of Chevalley groups

Elena Bunina, Pavel Gvozdevsky

Published 2023-11-03Version 1

In this paper we consider Chevalley groups over commutative rings with $1$, constructed by irreducible root systems of rank $>1$. We always suppose that for the systems $A_2, B_\ell, C_\ell, F_4, G_2$ our rings contain $1/2$ and for the system $G_2$ also $1/3$. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable. The same holds for adjoint Chevalley groups and for other Chevalley groups with some special condition of bounded generation. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself.