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Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations

Published online by Cambridge University Press:  20 November 2018

Jun Morita  and
Bertrand Rémy
Jun Morita
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan e-mail: morita@math.tsukuba.ac.jp
Bertrand Rémy
Affiliation:
Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France e-mail: remy@math.univ-lyon1.fr
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Abstract

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We prove simplicity for incomplete rank 2 Kac—Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).

Keywords

20G4420E4251E24Kac–Moody grouptwin treesimplicityroot systembuilding

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 390 - 400
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

The first author was supported in part by Grant-in-aid for Science Research (Monkasho Kakenhi) inJapan. The second author was supported in part by the Institut Universitaire de France.

References

[1] Abramenko, Peter and Rémy, Bertrand, Commensurators of some non-uniform tree lattices and Moufang twin trees. In: Essays in geometric group theory, Ramanujan Math. Soc. Lect. Notes Ser. 9, Mysore, 2009), 79–104.Google Scholar
[2] Benoist, Yves, Five lectures on lattices in semisimple Lie groups. Géométries á courbure négative ou nulle, groupes discrets et rigidités. Séminaires et Congrès 18, Société Mathématique de France, Paris, 2009, 117–176.Google Scholar
[3] Bessières, L., Parreau, A., and Bertrand, R. (eds.), Géométries `a courbure négative ou nulle, groupes discrets et rigidités. Séminaires et Congrès 18, Société Mathématique de France, Paris, 2009.Google Scholar
[4] Bourbaki, Nicolas, Lie groups and Lie algebras, Chapter 4–6. Actualités Scientifiques et Industrielles, Hermann, Paris, 1968, 1337.Google Scholar
[5] Bridson, Martin R. and André Haefliger, Metric spaces of non-positive curvature. Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin, 1999.Google Scholar
[6] Caprace, Pierre-Emmanuel and Rémy, Bertrand, Simplicity and superrigidity of twin building lattices. Invent. Math. 176 (2009), 169–221.http://dx.doi.org/10.1007/s00222-008-0162-6 Google Scholar
[7] Caprace, Pierre-Emmanuel and Rémy, Bertrand, Non-distortion of twin building lattices. Geom. Dedicata 147 (2010), 397–408.http://dx.doi.org/10.1007/s10711-010-9469-8 Google Scholar
[8] Caprace, Pierre-Emmanuel and Rémy, Bertrand, Simplicity of twin tree lattices with non-trivial commutation relations. Preprint of the Institut Camille Jordan 377, 2012.Google Scholar
[9] Meskin, Stephen, Nonresidually finite one-relator groups. Trans. Amer. Math. Soc. 164 (1972), 105–114.http://dx.doi.org/10.1090/S0002-9947-1972-0285589-5 Google Scholar
[10] Morita, Jun, Commutator relations in Kac–Moody groups. Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 21–22.http://dx.doi.org/10.3792/pjaa.63.21 Google Scholar
[11] Rémy, Bertrand, Construction de réseaux en théorie de Kac–Moody. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 475–478.http://dx.doi.org/10.1016/S0764-4442(00)80044-0 Google Scholar
[12] Rémy, Bertrand, Groupes de Kac–Moody déployés et presque déployés. Astérisque 277(2002).Google Scholar
[13] Rémy, Bertrand, Integrability of induction cocycles for Kac–Moody groups. Math. Ann. 333 (2005), 29–43.http://dx.doi.org/10.1007/s00208-005-0663-1 Google Scholar
[14] Rémy, Bertrand and Ronan, Mark A., Topological groups of Kac–Moody type, right-angled twinnings and their lattices. Comment. Math. Helv. 81 (2006), 191–219.http://dx.doi.org/10.4171/CMH/49 Google Scholar
[15] Ronan, Mark A. and Tits, Jacques, Twin trees. I. Invent. Math. 116 (1994), 463–479.http://dx.doi.org/10.1007/BF01231569 Google Scholar
[16] Ronan, Mark A. and Tits, Jacques, Twin trees. II. Local structure and a universal construction. Israel J. Math. 109 (1999), 349–377.http://dx.doi.org/10.1007/BF02775043 Google Scholar
[17] Rousseau, Guy, Euclidean buildings. In: Géométries `a courbure négative ou nulle, groupes discrets et rigidités. Séminaires et Congrès 18, Société Mathématique de France, Paris, 2009, 77–116.Google Scholar
[18] Tits, Jacques, On buildings and their applications. In: Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 1, Canad. Math. Congress, Montreal, QC, 1975, 209–220.Google Scholar
[19] Tits, Jacques, Uniqueness and presentation of Kac–Moody groups over fields. J. Algebra 105 (1987), 542–573.http://dx.doi.org/10.1016/0021-8693(87)90214-6 Google Scholar
[20] Tits, Jacques, Twin buildings and groups of Kac–Moody type. In: Groups, combinatorics & geometry (Durham, 1990), London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press, Cambridge, 1992, 249–286.Google Scholar
[21] Wilson, John S., Groups with every proper quotient finite. Proc. Cambridge Philos. Soc. 69 (1971), 373–391.http://dx.doi.org/10.1017/S0305004100046818 Google Scholar