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Commutators in pseudo-orthogonal groups. (English) Zbl 0847.20042
Given a pseudo-orthogonal group $G=O_{2n}R$ or $G=GO_{2n}R$ (this includes unitary, symplectic, orthogonal groups), the authors try to estimate the number $c(G)$ which by definition is such that any product of commutators in $G$ is also the product of at most $c(G)$ commutators. They succeed for semi-local rings $R$ and for rings satisfying a stable range condition that make the ring behave like a semi-local ring. There are several cases and the estimates vary from two to four. As expected, the proofs use clever matrix manipulations.
##### MSC:
 20G35 Linear algebraic groups over adèles and other rings and schemes 20F12 Commutator calculus (group theory) 20F05 Generators, relations, and presentations of groups 20H25 Other matrix groups over rings