Elementary equivalence of infinite-dimensional classical groups Original Research Article

Let D be a division ring such that the number of conjugacy classes of the multiplicative group D∗ is equal to the power of D∗. Suppose that H(V) is the group GL(V) or PGL(V), where V is a vector space of infinite dimension ϰ over D. We prove, in particular, that, uniformly in κ and D, the first-order theory of H(V) is mutually syntactically interpretable with the theory of the two-sorted structure 〈κ,D〉 (whose only relations are the division ring operations on D) in the second-order logic with quantification over arbitrary relations of power ⩽κ. A certain analogue of this results is proved for the groups View the MathML source and View the MathML source. These results imply criteria of elementary equivalence for infinite-dimensional classical groups of types View the MathML source, View the MathML source, GL, PGL over division rings, and solve, for these groups, a problem posed by U. Felgner. It follows from the criteria that if View the MathML source then View the MathML source and View the MathML source are second-order equivalent as sets.