Kneser-Tits hypothesis

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