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''Kneser–Tits conjecture''
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{{MSC|20G15|20G25}}
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A conjecture on the structure of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555501.png" />-simple simply-connected algebraic groups that are isotropic over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555502.png" />. Namely, the Kneser–Tits conjecture states that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555504.png" />-rational points of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555505.png" />-simple simply-connected isotropic [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555506.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555507.png" /> 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 [[#References|[1]]]. For a group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555508.png" /> (see [[Semi-simple algebraic group|Semi-simple algebraic group]]) the Kneser–Tits conjecture is equivalent to the Tannaka–Artin problem: Does the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k0555509.png" /> of elements of reduced norm one of a finite-dimensional skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555010.png" /> coincide with the commutator subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555011.png" /> of its multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555012.png" />? The Kneser–Tits conjecture has a close connection with questions of approximation in algebraic groups, rationality of group varieties and algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555013.png" />-theory.
 
  
The Kneser–Tits conjecture has been proved for the case of locally compact fields [[#References|[2]]] and also for global function fields [[#References|[3]]]. Moreover, for global fields of characteristic zero the method of descent in [[#References|[2]]] led to a proof of the Kneser–Tits conjecture for all algebraic groups except for those of types <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555015.png" />. However, the Kneser–Tits conjecture is not true in general, as follows from the negative solution to the Tannaka–Artin problem [[#References|[4]]]. As a result of this, progress has been made on the problem of the investigation of the measure of deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555016.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555017.png" />, which is expressible by the reduced [[Whitehead group|Whitehead group]] (cf. also [[Linear group|Linear group]]). The results obtained along these lines ([[#References|[5]]]–[[#References|[6]]]) form the basis of reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555018.png" />-theory. It has been proved in [[#References|[7]]] that the Kneser–Tits conjecture is also false for unitary groups, which in turn opens up a path for the development of reduced unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555019.png" />-theory.
 
  
====References====
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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
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Tits,  "Algebraic and abstract simple groups"  ''Ann. of Math.'' , '''80''' :  2  (1964)  pp. 313–329</TD></TR><TR><TD valign="top">[2]</TD> <TD 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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.P. Platonov,  "Arithmetic and structural problems in linear algebraic groups"  ''Transl. Amer. Math. Soc. (2)'' , '''109'''  (1977)  pp. 21–26  ''Proc. Internat. Congr. Mathematicians Vancouver''  (1974)  pp. 471–476</TD></TR><TR><TD valign="top">[4]</TD> <TD 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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.P. Platonov,  "The Tannaka–Artin problem and reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055550/k05555020.png" />-theory"  ''Math. USSR Izv.'' , '''40''' :  2  (1976)  pp. 211–244  ''Izv. Akad, Nauk SSSR. Ser. Mat.'' , '''40''' :  2  (1976)  pp. 227–261</TD></TR><TR><TD valign="top">[6]</TD> <TD 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</TD></TR><TR><TD valign="top">[7]</TD> <TD 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</TD></TR></table>
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[[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
 
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{{Cite|Ti}}. For a group of type $A_n$ (see
 
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[[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.
 
 
====Comments====
 
  
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The Kneser–Tits conjecture has been proved for the case of locally compact fields
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{{Cite|Pl}} and also for global function fields
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{{Cite|Pl2}}. Moreover, for global fields of characteristic zero the method of descent in
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{{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
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{{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
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[[Whitehead group|Whitehead group]] (cf. also
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[[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
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{{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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. PrasadM.S. Raghunathan,  "On the Kneser–Tits problem"  ''Math. Helv.'' , '''60'''  (1985)  pp. 107–121</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Tits,  "Groupes de Whitehead de groupes algébriques simples sur un corps (d'après V.P. Platonov et al.)" , ''Sem. Bourbaki Exp. 505 (1976/77)'' , ''Lect. notes in math.'' , '''677''' , Springer (1978)  pp. 218–236</TD></TR></table>
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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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|-
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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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|-
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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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|-
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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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|-
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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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|-
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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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|-
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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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Latest revision as of 18:52, 24 March 2012

2020 Mathematics Subject Classification: Primary: 20G15 Secondary: 20G25 [MSN][ZBL]


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 $G$ 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 [Ti]. For a group of type $A_n$ (see 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 [Pl] and also for global function fields [Pl2]. Moreover, for global fields of characteristic zero the method of descent in [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 [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 (cf. also Linear group). The results obtained along these lines ([Pl4][Pl5]) form the basis of reduced $K$-theory. It has been proved in [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

[Pl] 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
[Pl3] 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 MR0384857 MR0384858 Zbl 0333.20032
[Pl4] 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
[Pl5] 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
[PlYa] 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
[PrRa] G. Prasad, M.S. Raghunathan, "On the Kneser–Tits problem" Math. Helv., 60 (1985) pp. 107–121 MR0787664 Zbl 0574.20033
[Ti] J. Tits, "Algebraic and abstract simple groups" Ann. of Math., 80 : 2 (1964) pp. 313–329 MR0164968 Zbl 0131.26501
How to Cite This Entry:
Kneser-Tits hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kneser-Tits_hypothesis&oldid=13473
This article was adapted from an original article by V.I. Yanchevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article