Geodesic
$\mathrm{SL}_2$-factorisations of Chevalley groups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 22, Tome 394 (2011), pp. 20-32 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental $\mathrm{SL}_2$-factorizations of length $5N$, where $N$ is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov it follows that elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length $4N$. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to $3N$. On the other hand, for $\mathrm{SL}(n,R)$, over a Bezout ring $R$, we further improve the bound to $2N=n^2-n$. 
@article{ZNSL_2011_394_a1,
     author = {N. A. Vavilov and E. I. Kovach},
     title = {$\mathrm{SL}_2$-factorisations of {Chevalley} groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {20--32},
     year = {2011},
     volume = {394},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a1/}
}
TY  - JOUR
AU  - N. A. Vavilov
AU  - E. I. Kovach
TI  - $\mathrm{SL}_2$-factorisations of Chevalley groups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 20
EP  - 32
VL  - 394
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a1/
LA  - ru
ID  - ZNSL_2011_394_a1
ER  - 
%0 Journal Article
%A N. A. Vavilov
%A E. I. Kovach
%T $\mathrm{SL}_2$-factorisations of Chevalley groups
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 20-32
%V 394
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a1/
%G ru
%F ZNSL_2011_394_a1
N. A. Vavilov; E. I. Kovach. $\mathrm{SL}_2$-factorisations of Chevalley groups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 22, Tome 394 (2011), pp. 20-32. http://geodesic.mathdoc.fr/item/ZNSL_2011_394_a1/

[1] N. A. Vavilov, S. S. Sinchuk, “Razlozheniya tipa Dennisa–Vasershteina”, Zap. nauchn. semin. POMI, 375, 2010, 48–60 | MR | Zbl

[2] N. A. Vavilov, S. S. Sinchuk, “Parabolicheskie faktorizatsii rasschepimykh klassicheskikh grupp”, Algebra i Analiz, 23:4 (2011), 1–30

[3] N. A. Vavilov, A. V. Smolenskii, B. Suri, “Unitreugolnye faktorizatsii grupp Shevalle”, Zap. nauchn. semin. POMI, 388, 2011, 17–47 | MR

[4] R. Steinberg, Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[5] L. Babai, N. Nikolov, L. Pyber, “Product growth and mixing in finite groups”, 19th Annual ACM–SIAM Symposium on Discrete Algorithms, ACM–SIAM, 2008, 248–257 | MR | Zbl

[6] J. Bourgain, A. Gamburd, “Uniform expansion bounds for Cayley graphs of $\mathrm{SL}_2(\mathbb F_p)$”, Ann. Math., 167 (2008), 625–642 | DOI | MR | Zbl

[7] R. W. Carter, Simple groups of Lie type, Wiley, London et al., 1972 | MR | Zbl

[8] I. V. Erovenko, “$\mathrm{SL}_n(F[x])$ is not boundedly generated by elementary matrices: explicit proof”, Electronic J. Linear Algebra, 11 (2004), 162–167 | MR | Zbl

[9] D. R. Grayson, “$\mathrm{SK}_1$ of an interesting principal ideal domain”, J. Pure Appl. Algebra, 20 (1981), 157–163 | DOI | MR | Zbl

[10] H. A. Helfgott, “Growth and generation in $\mathrm{SL}_2(\mathbb Z/p\mathbb Z)$”, Ann. Math., 167 (2008), 601–623 | DOI | MR | Zbl

[11] F. Ischebeck, “Hauptidealringe mit nichttrivialer $\mathrm{SK}_1$-Gruppe”, Arch. Math., 35 (1980), 138–139 | DOI | MR | Zbl

[12] W. van der Kallen, “$\mathrm{SL}_3(\mathbb C[x])$ does not have bounded word length”, Springer Lect. Notes Math., 966, 1982, 357–361 | DOI | MR | Zbl

[13] M. Kassabov, A. Lubotzky, N. Nikolov, “Finite simple groups as expanders”, Proc. Nat. Acad. Sci. USA, 103 (2006), 6116–6119 | DOI | MR | Zbl

[14] M. Liebeck, N. Nikolov, A. Shalev, “A conjecture on product decompositions in simple groups”, Groups Geom. Dynamics, 4 (2010), 799–812 | DOI | MR | Zbl

[15] M. Liebeck, N. Nikolov, A. Shalev, “Groups of Lie type as products of $\mathrm{SL}_2$ subgroups”, J. Algebra, 326 (2011), 201–207 | DOI | MR | Zbl

[16] D. W. Morris, “Bounded generation of $\mathrm{SL}(n,A)$ (after D. Carter, G. Keller, and E. Paige)”, New York J. Math., 13 (2007), 383–421 | MR | Zbl

[17] N. Nikolov, “A product decomposition for the classical quasisimple groups”, J. Group Theory, 10 (2007), 43–53 | DOI | MR | Zbl

[18] N. Nikolov, L. Pyber, Product decomposition of quasirandom groups and a Jordan type theorem, 2007, arXiv: math/0703343 | MR

[19] S. Sinchuk, N. Vavilov, Parabolic factorisations of exceptional Chevalley groups (to appear)

[20] A. Stepanov, N. Vavilov, “On the length of commutators in Chevalley groups”, Israel Math. J., 185:1 (2011), 253–276 | DOI | MR | Zbl

[21] O. I. Tavgen, “Bounded generation of normal and twisted Chevalley groups over the rings of $S$-integers”, Contemp. Math., 131:1 (1992), 409–421 | DOI | MR | Zbl

[22] L. N. Vaserstein, “Polynomial parametrization for the solution of Diophantine equations and arithmetic groups”, Ann. Math., 171:2 (2010), 979–1009 | DOI | MR | Zbl

[23] N. Vavilov, “Structure of Chevalley groups over commutative rings”, Proc. Conf. Non-associative Algebras and Related Topics (Hiroshima – 1990), World Sci. Publ., London et al., 1991, 219–335 | MR | Zbl

[24] N. Vavilov, “A third look at weight diagrams”, Rendiconti Sem. Mat. Univ. Padova, 204:1 (2000), 201–250 | MR

[25] N. Vavilov, E. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations”, Acta Applicandae Math., 45 (1996), 73–115 | DOI | MR