Elementary equivalence of infinite-dimensional classical groups

✍ Scribed by Vladimir Tolstykh

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 363 KB
• Volume: 105
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(99)00050-0

✦ Synopsis

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 inÿnite dimension -over D. We prove, in particular, that, uniformly in -and D, the ÿrst-order theory of H (V ) is mutually syntactically interpretable with the theory of the two-sorted structure -; D (whose only relations are the division ring operations on D) in the second-order logic with quantiÿcation over arbitrary relations of power 6-. A certain analogue of this results is proved for the groups L(V ) and P L(V ). These results imply criteria of elementary equivalence for inÿnite-dimensional classical groups of types H = L, P L, 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 (V2) then -1 and -2 are second-order equivalent as sets.