Commutators in pseudo-orthogonal groups

Abstract

We study commutators in pseudo-orthogonal groups O 2n R (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO^R. We estimate the number of commutators, c(C>2nR) and c(G02 n R), needed to represent every element in the commutator subgroup. We show that c(O 2n R) < 4 if R satisfies the A-stable condition and either n > 3 or n = 2 and 1 is the sum of two units in R, and that c(GO 2n R) < 3 when the involution is trivial and A = R f . We also show that ciO^R) < 3 and c(GO 2n R) < 2 for the ordinary orthogonal group O 2rl R over a commutative ring R of absolute stable rank 1 where either n > 3 or n = 2 and 1 is the sum of two units in R.

Unitriangular factorizations of chevalley groups

“…We believe that similarly to [71,13] the commutator width of elementary Chevalley groups over a ring of stable rank 1 does not exceed 2 or 3.…”

“…• For symplectic group our Theorem 1 was established by You Hong [71]. For orthogonal groups a similar, but weaker result, under stronger stability conditions was established by Frank Arlinghaus, Leonid VAserstein and You Hong [13].…”

Commutator width of Chevalley groups over rings of stable rank 1

“…that every element of its commutator subgroup can be written as a product of two commutators. It was then generalised (with somewhat worse bounds) in [AVY95] to symplectic, orthogonal and unitary groups in even dimension in the context of hyperbolic unitary groups [BV00]. The goal of the present paper is to provide a similar result for exceptional groups.…”

“…Proof. For A and D this can be done by explicit matrix calculation (this is done for orthogonal group in [AVY95] and immediately follows for Spin group, since the central factor doesn't play any role). For E 7 it is not a good idea to write down matrices, but one can do something very similar.…”

“…The proof follows the line of those in [VW90,AVY95], but tries to avoid explicit matrix calculations, thus giving a simpler and (almost) uniform treatment for Chevalley groups of all normal types. Apart from exceptional groups, it also covers Spin and odd-dimensional orthogonal groups, which were not considered in the previous papers.…”