Regular bi-interpretability and finite axiomatizability of Chevalley groups

Abstract

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 (see Definition~2.2). The same holds for adjoint Chevalley groups and for bondedly generated Chevalley groups. 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.