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Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations

  • Jun Morita (a1) and Bertrand Rémy (a2)
Abstract

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).

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[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), 79104.
[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, 117176.
[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.
[4] Bourbaki, Nicolas, Lie groups and Lie algebras, Chapter 46. Actualités Scientifiques et Industrielles, Hermann, Paris, 1968, 1337.
[5] Bridson, Martin R. and André Haefliger, Metric spaces of non-positive curvature. Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin, 1999.
[6] Caprace, Pierre-Emmanuel and Rémy, Bertrand, Simplicity and superrigidity of twin building lattices. Invent. Math. 176 (2009), 169221.http://dx.doi.org/10.1007/s00222-008-0162-6
[7] Caprace, Pierre-Emmanuel and Rémy, Bertrand, Non-distortion of twin building lattices. Geom. Dedicata 147 (2010), 397408.http://dx.doi.org/10.1007/s10711-010-9469-8
[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.
[9] Meskin, Stephen, Nonresidually finite one-relator groups. Trans. Amer. Math. Soc. 164 (1972), 105114.http://dx.doi.org/10.1090/S0002-9947-1972-0285589-5
[10] Morita, Jun, Commutator relations in Kac–Moody groups. Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), 2122.http://dx.doi.org/10.3792/pjaa.63.21
[11] Rémy, Bertrand, Construction de réseaux en théorie de Kac–Moody. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 475478.http://dx.doi.org/10.1016/S0764-4442(00)80044-0
[12] Rémy, Bertrand, Groupes de Kac–Moody déployés et presque déployés. Astérisque 277(2002).
[13] Rémy, Bertrand, Integrability of induction cocycles for Kac–Moody groups. Math. Ann. 333 (2005), 2943.http://dx.doi.org/10.1007/s00208-005-0663-1
[14] Rémy, Bertrand and Ronan, Mark A., Topological groups of Kac–Moody type, right-angled twinnings and their lattices. Comment. Math. Helv. 81 (2006), 191219.http://dx.doi.org/10.4171/CMH/49
[15] Ronan, Mark A. and Tits, Jacques, Twin trees. I. Invent. Math. 116 (1994), 463479.http://dx.doi.org/10.1007/BF01231569
[16] Ronan, Mark A. and Tits, Jacques, Twin trees. II. Local structure and a universal construction. Israel J. Math. 109 (1999), 349377.http://dx.doi.org/10.1007/BF02775043
[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, 77116.
[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, 209220.
[19] Tits, Jacques, Uniqueness and presentation of Kac–Moody groups over fields. J. Algebra 105 (1987), 542573.http://dx.doi.org/10.1016/0021-8693(87)90214-6
[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, 249286.
[21] Wilson, John S., Groups with every proper quotient finite. Proc. Cambridge Philos. Soc. 69 (1971), 373391.http://dx.doi.org/10.1017/S0305004100046818
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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