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On the Kneser-Tits problem. (English) Zbl 0574.20033
The group \(SL_ 2(k)\) over a field \(k\) is generated by the elementary matrices \(\begin{pmatrix} 1&u \\ 0&1 \end{pmatrix}\) and \(\begin{pmatrix} 1&0 \\ v&1 \end{pmatrix}\), \(u\) and \(v\) in \(k\). For a \(k\)-algebraic group \(G\), let \(E_ G\) be the subgroup of \(G(k)\) generated by the \(k\)-unipotent elements of \(G\). The Kneser-Tits problem means: is \(E_ G=G(k)\) for any simply-connected, semi-simple, connected \(G\) with non-trivial \(k\)-unipotents? The authors give an affirmative answer for \(k\) a local field, a result already essentially proved by V. P. Platonov [Izv. Akad. Nauk SSSR, Ser. Mat. 33, 1211-1219 (1969; Zbl 0217.36301), ibid. 34, 775-777 (1970; Zbl 0236.20034)]. This result is a consequence of the reduction to the rank one: the answer is affirmative for \(k\) if and only if it is affirmative assuming moreover that the group has \(k\)-rank 1, and a consequence of the fact that there is no \(k\)-rank 1 form of exceptional groups for \(k\) local.
Reviewer: P.GĂ©rardin

20G25 Linear algebraic groups over local fields and their integers
20F05 Generators, relations, and presentations of groups
20G15 Linear algebraic groups over arbitrary fields
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