Elementary equivalence of infinite-dimensional classical groups

Abstract

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 ΓL(V) and PΓL(V) . These results imply criteria of elementary equivalence for infinite-dimensional classical groups of types H= ΓL , PΓL , GL, PGL over division rings, and solve, for these groups, a problem posed by U. Felgner. It follows from the criteria that if H(V 1)≡H(V 2) then κ 1 and κ 2 are second-order equivalent as sets.