Nikolai Gordeev and Ulf Rehmann: On Multicommutators for Simple Algebraic Groups,

Submission: 2000, Dec 12, published in J. Algebra, 245 (2001) 275--296

There are several examples of groups for which any pair of commutators can be written such that both of them have a common entry, and one can look for a similar property for $n$-tuples of commutators. We here answer, for simple algebraic groups over any field, the weaker question, under which condition the set of $n$-tuples of commutators with one common entry is Zariski dense in the set of all $n$-tuples of commutators. Surprisingly, there is a uniform bound on $n$ in terms of the so called Coxeter number of $G$ in order to answer the question positively. An analogoue result is proved for Lie algebras of simple and simply conncected algebraic groups.

2000 Mathematics Subject Classification: primary: 20G15; secondary: 20F12, 20E45

Keywords and Phrases: Groups, Commutators, Simple Algebraic Groups

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