Group theory
On the generation of discrete and topological Kac–Moody groups
[Sur les générateurs des groupes de Kac–Moody topologiques et discrets]
Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 695-699

This article shows that discrete or topological Kac–Moody groups defined over finite fields are in many cases 2-generated. We provide explicit bounds on the minimal number of generators for arbitrary Kac–Moody groups.

On montre que les groupes de Kac–Moody topologiques ou discrets définis sur des corps finis sont 2-engendrés dans de nombreux cas. On exhibe ensuite des bornes explicites sur le nombre minimal de générateurs pour un groupe de Kac–Moody arbitraire.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.03.009

Capdeboscq, Inna 1

1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
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Capdeboscq, Inna. On the generation of discrete and topological Kac–Moody groups. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 695-699. doi: 10.1016/j.crma.2015.03.009

[1] Abramenko, P.; Muhlherr, B. Presentations de certaines BN-paires jumeles comme sommes amalgames, C. R. Acad. Sci. Paris, Ser. I, Volume 325 (1997) no. 7, pp. 701-706

[2] Aschbacher, M.; Guralnick, R. Some applications of the first cohomology group, J. Algebra, Volume 90 (1984) no. 2, pp. 446-460

[3] Ben Messaoud, H. Almost split real forms for hyperbolic Kac–Moody Lie algebras, J. Phys. A, Volume 39 (2006) no. 44, pp. 13659-13690

[4] Capdeboscq, I. Bounded presentations of Kac–Moody groups, J. Group Theory, Volume 16 (2013) no. 6, pp. 899-905

[5] Capdeboscq, I.; Rémy, B. On some pro-p groups from infinite-dimensional Lie theory, Math. Z., Volume 278 (2014) no. 1–2, pp. 39-54

[6] Caprace, P.-E.; Rémy, B. Simplicité abstraite des groupes de Kac–Moody non-affines, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006) no. 8, pp. 539-544

[7] Carbone, L.; Chung, S.; Cobbs, L.; McRae, R.; Nandi, D.; Naqvi, Y.; Penta, D. Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, J. Phys. A, Volume 43 (2010) no. 15, p. 155209 ([30 p.])

[8] Carter, R. Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics, vol. 96, Cambridge University Press, Cambridge, UK, 2005

[9] Carter, R.W.; Chen, Y. Automorphisms of affine Kac–Moody groups and related Chevalley groups over rings, J. Algebra, Volume 155 (1993) no. 1, pp. 44-94

[10] Gorenstein, D.; Lyons, R.; Solomon, R. The Classification of the Finite Simple Groups, Number 1, American Mathematical Society Surveys and Monographs, vol. 40, 1998 (#3)

[11] Guralnick, R.; Kantor, W. Probabilistic generation of finite simple groups. Special issue in honor of Helmut Wielandt, J. Algebra, Volume 234 (2000) no. 2, pp. 743-792

[12] Hee, J.-Y. Construction de groupes tordus en théorie de Kac–Moody, C. R. Acad. Sci. Paris, Ser. I Math., Volume 310 (1990) no. 3, pp. 77-80

[13] Kantor, W.; Lubotzky, A. The probability of generating a finite classical group, Geom. Dedic., Volume 36 (1990) no. 1, pp. 67-87

[14] Menezes, N.; Quick, M.; Roney-Dougal, C. The probability of generating a finite simple group, Isr. J. Math., Volume 198 (2013), pp. 371-392

[15] Rousseau, G. Groupes de Kac–Moody déployés sur un corps local, II. Masures ordonnées, 2012 (preprint) | arXiv

[16] Tits, J. Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra, Volume 105 (1987), pp. 542-573

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