Elementary equivalence of infinite-dimensional classical groups

TL;DR

This paper establishes criteria for elementary equivalence of infinite-dimensional classical groups over division rings, linking their first-order theories to the second-order theory of their dimensions and division rings.

Contribution

It introduces a uniform interpretability of the first-order theories of classical groups with the second-order theory of their dimensions and division rings, solving a problem posed by Felgner.

Findings

Criteria for elementary equivalence of infinite-dimensional classical groups.

First-order theories are mutually interpretable with the second-order theory of dimensions and division rings.

Elementary equivalence implies second-order equivalence of the dimension sets.

Abstract

Let D be a division ring such that the number of conjugacy classes in 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 an infinite-dimensional vector space over D. We prove, in particular, that, uniformly in dim(V) and D, the first-order theory of H(V) is mutually syntactically interpretable with the theory of the two-sorted structure <dim(V),D> (whose only relations are the division ring operations on D) in the second-order logic with quantification over arbitrary relations of power <= dim(V). A certain analogue of this results is proved for the groups the collinear groups GammaL(V) and PGammaL(V). These results imply criteria of elementary equivalence for infinite-dimensional classical groups of types H=GammaL, PGammaL, GL, PGL over division rings, and solve, for these groups, a problem posed by Felgner. It follows from the criteria that if H(V_1), H(V_2) are elementarily equivalent, then the cardinals dim(V_1) and dim(V_2) are second order equivalent as sets.