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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.
20G35Linear algebraic groups over adèles and other rings and schemes
20F12Commutator calculus (group theory)
20F05Generators, relations, and presentations of groups
20H25Other matrix groups over rings