View source for Kneser-Tits hypothesis

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{{MSC|20G15|20G25}}


The ''Kneser–Tits conjecture''
is a conjecture on the structure of the $k$-simple simply-connected algebraic groups that are isotropic over a field $k$. Namely, the Kneser–Tits conjecture states that the group $G_k$ of $k$-rational points of a $k$-simple simply-connected isotropic
[[Algebraic group|algebraic group]] $G$ over a field $k$ is generated by its unipotent elements. This conjecture was stated in a somewhat less general form by M. Kneser; the general statement is due to J. Tits
{{Cite|Ti}}. For a group of type $A_n$ (see
[[Semi-simple algebraic group|Semi-simple algebraic group]]) the Kneser–Tits conjecture is equivalent to the Tannaka–Artin problem: Does the group $\def\SL{\textrm{SL}}\SL(1,D)$ of elements of reduced norm one of a finite-dimensional skew-field $D$ coincide with the commutator subgroup $[D^*,D^*]$ of its multiplicative group $D^*$? The Kneser–Tits conjecture has a close connection with questions of approximation in algebraic groups, rationality of group varieties and algebraic $K$-theory.

The Kneser–Tits conjecture has been proved for the case of locally compact fields
{{Cite|Pl}} and also for global function fields
{{Cite|Pl2}}. Moreover, for global fields of characteristic zero the method of descent in
{{Cite|Pl}} led to a proof of the Kneser–Tits conjecture for all algebraic groups except for those of types $E_6$ and $E_8$. However, the Kneser–Tits conjecture is not true in general, as follows from the negative solution to the Tannaka–Artin problem
{{Cite|Pl3}}. As a result of this, progress has been made on the problem of the investigation of the measure of deviation of $\SL(1,D)$ from $[D^*,D^*]$, which is expressible by the reduced
[[Whitehead group|Whitehead group]] (cf. also
[[Linear group|Linear group]]). The results obtained along these lines ({{Cite|Pl4}}–{{Cite|Pl5}}) form the basis of reduced $K$-theory. It has been proved in
{{Cite|PlYa}} that the Kneser–Tits conjecture is also false for unitary groups, which in turn opens up a path for the development of reduced unitary $K$-theory.

====References====
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|valign="top"|{{Ref|Pl}}||valign="top"|  V.P. Platonov,   "The problem of strong approximation and the Kneser–Tits conjecture for algebraic groups"  ''Math. USSR Izv.'', '''3''' :  6  (1969)  pp. 1135–1148  ''Izv. Akad. Nauk SSSR. Ser. Mat.'', '''33''' :  6  (1969)  pp. 1121–1220   {{ZBL|0217.36301}} 
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|valign="top"|{{Ref|Pl3}}||valign="top"|  V.P. Platonov,   "On the Tannaka–Artin problem"  ''Soviet Math. Dokl.'', '''16'''  (1975)  pp. 468–473  ''Dokl. Akad. Nauk SSSR'', '''221''' :  5  (1975)  pp. 1038–1041  {{MR|0384857}} {{MR|0384858}}  {{ZBL|0333.20032}} 
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|valign="top"|{{Ref|Pl4}}||valign="top"|  V.P. Platonov,   "The Tannaka–Artin problem and reduced $K$-theory"  ''Math. USSR Izv.'', '''40''' :  2  (1976)  pp. 211–244  ''Izv. Akad, Nauk SSSR. Ser. Mat.'', '''40''' :  2  (1976)  pp. 227–261   
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|valign="top"|{{Ref|Pl5}}||valign="top"|  V.P. Platonov,   "The infinitude of the reduced Whitehead group in the Tannaka–Artin problem"  ''Math. USSR Sb.'', '''29'''  (1976)  pp. 167–176  ''Mat. Sb.'', '''100''' :  2  (1976)  pp. 191–200   
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|valign="top"|{{Ref|PlYa}}||valign="top"|  V.P. Platonov,   V.I. Yanchevskii,   "On the Kneser–Tits conjecture for unitary groups"  ''Soviet Math. Dokl.'', '''16'''  (1975)  pp. 1456–1460  ''Dokl. Akad. Nauk SSSR'', '''225''' :  1  (1975)  pp. 48–51   {{ZBL|0343.16016}} 
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|valign="top"|{{Ref|PrRa}}||valign="top"|  G. Prasad,   M.S. Raghunathan,   "On the Kneser–Tits problem"  ''Math. Helv.'', '''60'''  (1985)  pp. 107–121  {{MR|0787664}}  {{ZBL|0574.20033}} 
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|valign="top"|{{Ref|Ti}}||valign="top"|  J. Tits,   "Algebraic and abstract simple groups"  ''Ann. of Math.'', '''80''' :  2  (1964)  pp. 313–329  {{MR|0164968}}  {{ZBL|0131.26501}} 
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