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
Y. Barnea, M. Ershov and T. Weigel, Abstract commensurators of profinite groups, Transactions of the American Mathematical Society 363 (2011), 5381–5417.
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.
M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer, Berlin, 1999.
M. Burger and S. Mozes, Lattices in product of trees, Publications Mathématiques. Institut de Hautes études Scientifiques (2000), pno. 92, 151–194 (2001).
A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Annals of Mathematics. Second Series 72 (1960), 179–188.
A. Borel, Density and maximality of arithmetic subgroups, Journal für die Reine und Angewandte Mathematik 224 (1966), 78–89.
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.
N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971.
U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups, Inventiones Mathematicae 163 (2006), 415–454.
A. Borel and J. Tits, Homomorphismes “abstraits” de groupes algébriques simples, Annals of Mathematics. Second Series 97 (1973), 499–571.
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.
L. Carbone and H. Garland, Lattices in Kac-Moody groups, Mathematical Research Letters 6 (1999), 439–448.
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.
P.-E. Caprace and N. Monod, Some properties of non-positively curved lattices, Comptes Rendus Mathématique. Académie des Sciences. Paris 346 (2008), 857–862.
P.-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: structure theory, Journal of Topology 2 (2009), 661–700
P.-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: discrete subgroups, Journal of Topology 2 (2009), 701–746
P.-E. Caprace and N. Monod, Fixed points and amenability in non-positive curvature, in preparation.
P.-E. Caprace and B. Rémy, Simplicity and superrigidity of twin building lattices, Inventiones Mathematicae 176 (2009), 169–221.
D. I. Cartwright and T. Steger, A family of à n-groups, Israel Journal of Mathematics 103 (1998), 125–140.
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.
J. Dymara and T. Januszkiewicz, Cohomology of buildings and of their automorphism groups, Inventiones Mathematicae 150 (2002), 579–627.
G. Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, Journal für die Reine und Angewandte Mathematik 274/275 (1975), 125–138.
B. Kleiner, The local structure of length spaces with curvature bounded above, Mathematische Zeitschrift 231 (1999), 409–456.
A. I. Mal’cev, On isomorphic matrix representations of infinite groups, Rec. Math. (Mat. Sbornik) 8(50) (1940), 405–422.
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 17, Springer-Verlag, Berlin, 1991.
O. Mathieu, Construction d’un groupe de Kac-Moody et applications, Compositio Mathematica 69 (1989), 37–60 (French).
I. Mineyev, N. Monod and Y. Shalom, Ideal bicombings for hyperbolic groups and applications, Topology 43 (2004), 1319–1344.
N. Monod, Superrigidity for irreducible lattices and geometric splitting, Journal of the American Mathematical Society 19 (2006), 781–814.
G. Prasad, Strong approximation for semi-simple groups over function fields, Annals of Mathematics. Second Series 105 (1977), 553–572.
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.
M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York, 1972.
B. Rémy, Construction de réseaux en théorie de Kac-Moody, Comptes Rendus Mathématique. Académie des Sciences. Paris 329 (1999), 475–478.
B. Rémy, Groupes de Kac-Moody déployés et presque déployés, Astérisque 277 (2002) (French).
J. Tits, Algebraic and abstract simple groups, Annals of Mathematics. Second Series 80 (1964), 313–329.
J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin, 1974.
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.
J. Tits, Uniqueness and presentation of Kac-Moody groups over fields, Journal of Algebra 105 (1987), 542–573.
J. S. Wilson, Groups with every proper quotient finite, Proceedings of the Cambridge Philosophical Society 69 (1971), 373–391.
K. Wortman, Quasi-isometric rigidity of higher rank S-arithmetic lattices, Geometry and Topology 11 (2007), 995–1048.
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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