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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.Gérardin

##### MSC:
 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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