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Buildings and Kac-Moody Groups

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Buildings, Finite Geometries and Groups

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 10))

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Abstract

This survey paper provides an overview of some aspects of the theory buildings in connection with geometric and analytic group theory.

Subject Classifications: AMS classification (2000): 20E42, 51E24, 20F32, 20F67, 20F69, 22F, 22F10, 22F50

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References

  1. Peter Abramenko and Kenneth S. Brown, Buildings, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008, Theory and applications. MR2439729 (2009g:20055)

    Google Scholar 

  2. Peter Abramenko, Finiteness properties of groups acting on twin buildings, Groups: topological, combinatorial and arithmetic aspects, London Math. Soc. Lecture Note Ser., vol. 311, Cambridge Univ. Press, Cambridge, 2004, pp. 21–26. MR2073344

    Google Scholar 

  3. Peter Abramenko and Bernhard Mühlherr, Présentations de certaines BN-paires jumelées comme sommes amalgamées, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 7, 701–706. MR1483702 (98h:20043)

    Google Scholar 

  4. Peter Abramenko and Bertrand Rémy, Commensurators of some non-uniform tree lattices and Moufang twin trees, Essays in geometric group theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 9, Ramanujan Math. Soc., Mysore, 2009, pp. 79–104. MR2605356

    Google Scholar 

  5. Mladen Bestvina and Koji Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (2009), no. 1, 11–40. MR2507218 (2010m:53060)

    Google Scholar 

  6. Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran Šuniḱ, Branch groups, Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 989–1112. MR2035113 (2005f:20046)

    Google Scholar 

  7. Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer, Berlin, 1999. MR1744486 (2000k:53038)

    Google Scholar 

  8. Marc Burger and Shahar Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000), no. 92, 151–194 (2001). MR1839489 (2002i:20042)

    Google Scholar 

  9. Marc Burger and Nicolas Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002), no. 2, 219–280. MR1911660 (2003d:53065a)

    Google Scholar 

  10. Nicolas Bourbaki, Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles, No. 1306, Hermann, Paris, 1963. MR0179291 (31 #3539)

    Google Scholar 

  11. Nicolas 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. MR0240238 (39 #1590)

    Google Scholar 

  12. Nicolas Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. MR0358652 (50 #11111)

    Google Scholar 

  13. Marc Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997), no. 2, 245–268. MR1445387 (98c:20056)

    Google Scholar 

  14. Marc Bourdon and Hervé Pajot, Rigidity of quasi-isometries for some hyperbolic buildings, Comment. Math. Helv. 75 (2000), no. 4, 701–736. MR1789183 (2003a:30027)

    Google Scholar 

  15. Udo Baumgartner, Jacqui Ramagge, and Bertrand Rémy, Contraction groups in complete Kac-Moody groups, Groups Geom. Dyn. 2 (2008), no. 3, 337–352. MR2415303 (2009g:22014)

    Google Scholar 

  16. Udo Baumgartner, Bertrand Rémy, and George A. Willis, Flat rank of automorphism groups of buildings, Transform. Groups 12 (2007), no. 3, 413–436. MR2356316 (2008j:22035)

    Google Scholar 

  17. Uri Bader and Yehuda Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math. 163 (2006), no. 2, 415–454. MR2207022 (2006m:22017)

    Google Scholar 

  18. Fran çois Bruhat and Jacques Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. (1972), no. 41, 5–251. MR0327923 (48 #6265)

    Google Scholar 

  19. Fran çois Bruhat and Jacques Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. (1984), no. 60, 197–376. MR756316 (86c:20042)

    Google Scholar 

  20. Pierre-Emmanuel Caprace, “Abstract” homomorphisms of split Kac-Moody groups, Mem. Amer. Math. Soc. 198 (2009), no. 924, xvi+84. MR2499773 (2010d:20057)

    Google Scholar 

  21. Lisa Carbone, Mikhail Ershov, and Gordon Ritter, Abstract simplicity of complete Kac-Moody groups over finite fields, J. Pure Appl. Algebra 212 (2008), no. 10, 2147–2162. MR2418160 (2009d:20067)

    Google Scholar 

  22. Pierre-Emmanuel Caprace and Koji Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (2010), no. 5, 1296–1319. MR2585575

    Google Scholar 

  23. Lisa Carbone and Howard Garland, Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math. 5 (2003), no. 5, 813–867. MR2017720 (2004m:17031)

    Google Scholar 

  24. Pierre-Emmanuel Caprace and Frédéric Haglund, On geometric flats in the CAT(0) realization of Coxeter groups and Tits buildings, Canad. J. Math. 61 (2009), no. 4, 740–761. MR2541383 (2010k:20051)

    Google Scholar 

  25. Pierre-Emmanuel Caprace and Jean Lécureux, Combinatorial and group-theoretic compactifications of buildings, to appear in the Annales de l’Institut Fourier

    Google Scholar 

  26. Pierre-Emmanuel Caprace and Bertrand Rémy, Groups with a root group datum, Innov. Incidence Geom. 9 (2009), 5–77. MR2658894

    Google Scholar 

  27. Pierre-Emmanuel Caprace and Bertrand Rémy, Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009), no. 1, 169–221. MR2485882 (2010d:20056)

    Google Scholar 

  28. Pierre-Emmanuel Caprace and Bertrand Rémy, Non-distortion of twin building lattices, Geom. Dedicata 147 (2010), 397–408. MR2660586

    Google Scholar 

  29. Michael W. Davis, Buildings are CAT (0), Geometry and cohomology in group theory (Durham, 1994), London Math. Soc. Lecture Note Ser., vol. 252, Cambridge Univ. Press, Cambridge, 1998, pp. 108–123. MR1709955 (2000i:20068)

    Google Scholar 

  30. Jan Dymara and Tadeusz Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math. 150 (2002), no. 3, 579–627. MR1946553 (2003j:20052)

    Google Scholar 

  31. Mikhail Ershov, Golod-Shafarevich groups with property (T) and Kac-Moody groups, Duke Math. J. 145 (2008), no. 2, 309–339. MR2449949 (2009i:20060)

    Google Scholar 

  32. Hillel Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR0146298 (26 #3820)

    Google Scholar 

  33. Ralf Gramlich, Max Horn, and Bernhard Mühlherr, Abstract involutions of algebraic groups and of Kac-Moody groups, to appear in J. Group Theory.

    Google Scholar 

  34. Étienne Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no. 3-4, 329–407. MR1876932 (2003a:37032)

    Google Scholar 

  35. Ralf Gramlich and Bernhard Mühlherr, Lattices from involutions of Kac-Moody groups, in Abstracts from the mini-workshop held November 16–22, 2008, organized by S. Goodwin and R. Gramlich, Oberwolfach Reports. Vol. 5, no. 4.

    Google Scholar 

  36. Yves Guivarc’h and Bertrand Rémy, Group-theoretic compactification of Bruhat-Tits buildings, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 871–920. MR2316977 (2008f:20056)

    Google Scholar 

  37. Günter Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, Invent. Math. 7 (1969), 33–54. MR0284441 (44 #1667)

    Google Scholar 

  38. Frédéric Haglund and Frédéric Paulin, Simplicité de groupes d’automorphismes d’espaces à courbure négative, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 181–248 (electronic). MR1668359 (2000b:20034)

    Google Scholar 

  39. Bruce Kleiner and Bernhard Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997), no. 86, 115–197 (1998). MR1608566 (98m:53068)

    Google Scholar 

  40. Jean Lécureux, Amenability of actions on the boundary of a building, Int. Math. Res. Not. IMRN (2010), no. 17, 3265–3302. MR2680274

    Google Scholar 

  41. Jean Lécureux, Hyperbolic configurations of roots and Hecke algebras, J. Algebra 323 (2010), no. 5, 1454–1467. MR2584964

    Google Scholar 

  42. Ian G. Macdonald, Spherical functions on a group of p-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, 1971, Publications of the Ramanujan Institute, No. 2. MR0435301 (55 #8261)

    Google Scholar 

  43. Wilhelm Magnus, Residually finite groups, Bull. Amer. Math. Soc. 75 (1969), 305–316. MR0241525 (39 #2865)

    Google Scholar 

  44. Gregori A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 17, Springer, Berlin, 1991. MR1090825 (92h:22021)

    Google Scholar 

  45. Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer, Berlin, 1988. MR959135 (90a:30132)

    Google Scholar 

  46. Calvin C. Moore, Compactifications of symmetric spaces, Amer. J. Math. 86 (1964), 201–218. MR0161942 (28 #5146)

    Google Scholar 

  47. Nicolas Monod and Yehuda Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom. 67 (2004), no. 3, 395455. MR2153026 (2006g:53051)

    Google Scholar 

  48. Gregori A. Margulis and Èrnest B. Vinberg, Some linear groups virtually having a free quotient, J. Lie Theory 10 (2000), no. 1, 171–180. MR1748082 (2001h:22016)

    Google Scholar 

  49. Guennadi A. Noskov and Èrnest B. Vinberg, Strong Tits alternative for subgroups of Coxeter groups, J. Lie Theory 12 (2002), no. 1, 259–264. MR1885045 (2002k:20072)

    Google Scholar 

  50. Gopal Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. (1989), no. 69, 91–117, With an appendix by Moshe Jarden and the author. MR1019962 (91c:22023)

    Google Scholar 

  51. Diego Rattaggi, A finitely presented torsion-free simple group, J. Group Theory 10 (2007), no. 3, 363–371. MR2320973 (2008c:20055)

    Google Scholar 

  52. Bertrand Rémy, Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 6, 475–478. MR1715140 (2001d:20028)

    Google Scholar 

  53. Bertrand Rémy, Classical and non-linearity properties of Kac-Moody lattices, Rigidity in dynamics and geometry (Cambridge, 2000), Springer, Berlin, 2002, pp. 391–406. MR1919413 (2003g:20071)

    Google Scholar 

  54. Bertrand Rémy, Groupes de Kac-Moody déployés et presque déployés, Astérisque (2002), no. 277, viii + 348. MR1909671 (2003d:20036)

    Google Scholar 

  55. Bertrand Rémy, Immeubles de Kac-Moody hyperboliques, groupes non isomorphes de même immeuble, Geom. Dedicata 90 (2002), 29–44. MR1898149 (2003c:20033)

    Google Scholar 

  56. Bertrand Rémy, Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups, Geom. Funct. Anal. 14 (2004), no. 4, 810–852, With an appendix by P. Bonvin. MR2084981 (2005g:22024)

    Google Scholar 

  57. Bertrand Rémy, Integrability of induction cocycles for Kac-Moody groups, Math. Ann. 333 (2005), no. 1, 29–43. MR2169827 (2006k:22018)

    Google Scholar 

  58. Bertrand Rémy, Kac-Moody groups as discrete groups, Essays in geometric group theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 9, Ramanujan Math. Soc., Mysore, 2009, pp.105–124. MR2605357

    Google Scholar 

  59. Guy Rousseau, Euclidean buildings, Géométries à courbure négative ou nulle, groupes discrets et rigidités (Paris), Sémin. Congr., vol. 18, Soc. Math. France, 2009, pp. 77–116. MR2655310

    Google Scholar 

  60. Bertrand Rémy and Mark Ronan, Topological groups of Kac-Moody type, right-angled twinnings and their lattices, Comment. Math. Helv. 81 (2006), no. 1, 191–219. MR2208804 (2007b:20063)

    Google Scholar 

  61. Ichirô Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. (2) 71 (1960), 77–110. MR0118775 (22 #9546)

    Google Scholar 

  62. Ichirô Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Inst. Hautes Études Sci. Publ. Math. (1963), no. 18, 5–69. MR0195863 (33 #4059)

    Google Scholar 

  63. Yehuda Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), no. 1, 1–54. MR1767270 (2001k:22022)

    Google Scholar 

  64. Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968, Notes prepared by John Faulkner and Robert Wilson. MR0466335 (57 #6215)

    Google Scholar 

  65. Anne Thomas, Lattices acting on right-angled buildings, Algebr. Geom. Topol. 6 (2006), 1215–1238. MR2253444 (2008b:22003)

    Google Scholar 

  66. Anne Thomas, On the set of covolumes of lattices for Fuchsian buildings, C. R. Math. Acad. Sci. Paris 344 (2007), no. 4, 215–218. MR2292989 (2008d:20053)

    Google Scholar 

  67. Jacques Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), no. 2, 542–573. MR873684 (89b:17020)

    Google Scholar 

  68. Jacques Tits, Groupes associés aux algèbres de Kac-Moody, Astérisque (1989), no. 177-178, Exp. No. 700, 7–31, Séminaire Bourbaki, Vol. 1988/89. MR1040566 (91c:22034)

    Google Scholar 

  69. André Weil, L’intégration dans les groupes topologiques et ses applications, Actual. Sci. Ind., no. 869, Hermann, 1940. MR0005741 (3,198b)

    Google Scholar 

  70. George Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), no. 2, 341–363. MR1299067 (95j:22010)

    Google Scholar 

  71. Xiangdong Xie, Quasi-isometric rigidity of Fuchsian buildings, Topology 45 (2006), no. 1, 101–169. MR2170496 (2006m:53064)

    Google Scholar 

  72. Efim Zelmanov, Infinite algebras and pro-p groups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 403–413. MR2195460 (2006k:20053)

    Google Scholar 

  73. Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser, Basel, 1984. MR776417 (86j:22014)

    Google Scholar 

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Rémy, B. (2012). Buildings and Kac-Moody Groups. In: Sastry, N. (eds) Buildings, Finite Geometries and Groups. Springer Proceedings in Mathematics, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0709-6_11

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