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A lattice in more than two Kac-Moody groups is arithmetic

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Abstract

Let Γ < G 1 × … × G n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.

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References

  1. Y. Barnea, M. Ershov and T. Weigel, Abstract commensurators of profinite groups, Transactions of the American Mathematical Society 363 (2011), 5381–5417.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Borel and G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, Journal für die Reine und Angewandte Mathematik 298 (1978), 53–64.

    MATH  MathSciNet  Google Scholar 

  3. M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer, Berlin, 1999.

    MATH  Google Scholar 

  4. M. Burger and S. Mozes, Lattices in product of trees, Publications Mathématiques. Institut de Hautes études Scientifiques (2000), pno. 92, 151–194 (2001).

  5. A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Annals of Mathematics. Second Series 72 (1960), 179–188.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Borel, Density and maximality of arithmetic subgroups, Journal für die Reine und Angewandte Mathematik 224 (1966), 78–89.

    MATH  MathSciNet  Google Scholar 

  7. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.

    Google Scholar 

  8. N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971.

    Google Scholar 

  9. U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups, Inventiones Mathematicae 163 (2006), 415–454.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Borel and J. Tits, Homomorphismes “abstraits” de groupes algébriques simples, Annals of Mathematics. Second Series 97 (1973), 499–571.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Carbone, M. Ershov and G. Ritter, Abstract simplicity of complete Kac-Moody groups over finite fields, Journal of Pure and Applied Algebra 212 (2008), 2147–2162.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. Carbone and H. Garland, Lattices in Kac-Moody groups, Mathematical Research Letters 6 (1999), 439–448.

    MATH  MathSciNet  Google Scholar 

  13. P.-E. Caprace and F. Haglund, On geometric flats in the CAT(0) realization of Coxeter groups and Tits buildings, Canadian Journal of Mathematics 61 (2009), 740–761.

    Article  MATH  MathSciNet  Google Scholar 

  14. P.-E. Caprace and N. Monod, Some properties of non-positively curved lattices, Comptes Rendus Mathématique. Académie des Sciences. Paris 346 (2008), 857–862.

    Article  MATH  MathSciNet  Google Scholar 

  15. P.-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: structure theory, Journal of Topology 2 (2009), 661–700

    Article  MATH  MathSciNet  Google Scholar 

  16. P.-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: discrete subgroups, Journal of Topology 2 (2009), 701–746

    Article  MATH  MathSciNet  Google Scholar 

  17. P.-E. Caprace and N. Monod, Fixed points and amenability in non-positive curvature, in preparation.

  18. P.-E. Caprace and B. Rémy, Simplicity and superrigidity of twin building lattices, Inventiones Mathematicae 176 (2009), 169–221.

    Article  MATH  MathSciNet  Google Scholar 

  19. D. I. Cartwright and T. Steger, A family of à n-groups, Israel Journal of Mathematics 103 (1998), 125–140.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Davis, Buildings are CAT(0), in Geometry and Cohomology in Group Theory (Durham, 1994), London Mathematical Society Lecture Note Series, Vol. 252, Cambridge University Press, 1998, pp. 108–123.

  21. J. Dymara and T. Januszkiewicz, Cohomology of buildings and of their automorphism groups, Inventiones Mathematicae 150 (2002), 579–627.

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, Journal für die Reine und Angewandte Mathematik 274/275 (1975), 125–138.

    MathSciNet  Google Scholar 

  23. B. Kleiner, The local structure of length spaces with curvature bounded above, Mathematische Zeitschrift 231 (1999), 409–456.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. I. Mal’cev, On isomorphic matrix representations of infinite groups, Rec. Math. (Mat. Sbornik) 8(50) (1940), 405–422.

    MathSciNet  Google Scholar 

  25. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 17, Springer-Verlag, Berlin, 1991.

    Book  MATH  Google Scholar 

  26. O. Mathieu, Construction d’un groupe de Kac-Moody et applications, Compositio Mathematica 69 (1989), 37–60 (French).

    MATH  MathSciNet  Google Scholar 

  27. I. Mineyev, N. Monod and Y. Shalom, Ideal bicombings for hyperbolic groups and applications, Topology 43 (2004), 1319–1344.

    Article  MATH  MathSciNet  Google Scholar 

  28. N. Monod, Superrigidity for irreducible lattices and geometric splitting, Journal of the American Mathematical Society 19 (2006), 781–814.

    Article  MATH  MathSciNet  Google Scholar 

  29. G. Prasad, Strong approximation for semi-simple groups over function fields, Annals of Mathematics. Second Series 105 (1977), 553–572.

    Article  MATH  MathSciNet  Google Scholar 

  30. G. Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bulletin de la Société Mathématique de France 110 (1982), 197–202.

    MATH  Google Scholar 

  31. M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York, 1972.

    MATH  Google Scholar 

  32. B. Rémy, Construction de réseaux en théorie de Kac-Moody, Comptes Rendus Mathématique. Académie des Sciences. Paris 329 (1999), 475–478.

    Article  MATH  Google Scholar 

  33. B. Rémy, Groupes de Kac-Moody déployés et presque déployés, Astérisque 277 (2002) (French).

  34. J. Tits, Algebraic and abstract simple groups, Annals of Mathematics. Second Series 80 (1964), 313–329.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin, 1974.

    MATH  Google Scholar 

  36. J. Tits, Groups and group functors attached to Kac-Moody data, (Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984), Lecture Notes in Mathematics, Vol. 1111, Springer-Verlag, Berlin, 1985, pp. 193–223.

    Google Scholar 

  37. J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, Journal of Algebra 105 (1987), 542–573.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. S. Wilson, Groups with every proper quotient finite, Proceedings of the Cambridge Philosophical Society 69 (1971), 373–391.

    Article  MATH  Google Scholar 

  39. K. Wortman, Quasi-isometric rigidity of higher rank S-arithmetic lattices, Geometry and Topology 11 (2007), 995–1048.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. UCLouvain, 1348, Louvain-la-Neuve, Belgium

    Pierre-Emmanuel Caprace

  2. EPFL, 1015, Lausanne, Switzerland

    Nicolas Monod

Authors
  1. Pierre-Emmanuel Caprace
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  2. Nicolas Monod
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Correspondence to Pierre-Emmanuel Caprace.

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F.N.R.S. Research Associate.

Supported in part by the Swiss National Science Foundation.

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Caprace, PE., Monod, N. A lattice in more than two Kac-Moody groups is arithmetic. Isr. J. Math. 190, 413–444 (2012). https://doi.org/10.1007/s11856-012-0006-3

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  • DOI: https://doi.org/10.1007/s11856-012-0006-3

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