Complete first-order theories of some classical matrix groups over algebraic integers

Abstract

Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SLn(O), the general linear group GLn(O) and the subgroup Tn(O) of GLn(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.