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Basic Definitions and Properties

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Finiteness Properties of Arithmetic Groups Acting on Twin Buildings

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2109))

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Abstract

In this first chapter we introduce notions and statements that will be needed later on and that are more or less generally known. The focus is on developing the necessary ideas in their natural context, proofs are generally omitted. For the reader who is interested in more details, an effort has been made to give plenty of references. Where less appropriate references are known to the author, the exposition is more detailed.

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References

  1. Abramenko, P., Brown, K.S.: Buildings: Theory and Applications. Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)

    Google Scholar 

  2. Abramenko, P.: Twin Buildings and Applications to S-Arithmetic Groups. Lecture Notes in Mathematics, vol. 1641. Springer, New York (1996)

    Google Scholar 

  3. Abramenko, P., Ronan, M.: A characterization of twin buildings by twin apartments. Geom. Dedicata 73, 1–9 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Artin, E.: Algebraic Numbers and Algebraic Functions. Gordon and Breach, London (1967)

    MATH  Google Scholar 

  5. Abramenko, P., van Maldeghem, H.: 1-twinnings of buildings. Math. Z. 238, 187–203 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ballmann, W.: Lectures on Spaces of Nonpositive Curvature. DMV Seminar, vol. 25. Birkhäuser, Basel (1995)

    Google Scholar 

  7. Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bestvina, M.: PL Morse theory. Math. Commun. 13, 149–162 (2008)

    MATH  MathSciNet  Google Scholar 

  9. Bux, K.U., Gramlich, R., Witzel, S.: Finiteness properties of Chevalley groups over a polynomial ring over a finite field (2010). Preprint [ArXiv:0908.4531v5]

    Google Scholar 

  10. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Die Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)

    Google Scholar 

  11. Bieri, R.: Homological Dimension of Discrete Groups. Mathematics Department, Queen Mary College, London (1976)

    MATH  Google Scholar 

  12. Bux, K.U., Köhl, R., Witzel, S.: Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem. Ann. Math. (2) 177, 311–366 (2013)

    Google Scholar 

  13. Borel, A.: Linear algebraic groups. In: Algebraic Groups and Discontinuous Subgroups. Proceedings of the Symposium on Pure Mathematics, Boulder, 1965, pp. 3–19. American Mathematical Society, Providence (1966)

    Google Scholar 

  14. Borel, A.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 126. Springer, Berlin (1991)

    Google Scholar 

  15. Bourbaki, N.: Lie Groups and Lie Algebras, Chaps. 4–6. Elements of Mathematics. Springer, Berlin (2002)

  16. Bredon, G.E.: Topology and Geometry. Graduate Texts in Mathematics, vol. 139. Springer, Berlin (1993)

    Google Scholar 

  17. Bridson, M.R.: Geodesics and curvature in metric simplicial complexes. In: Group Theory from a Geometrical Viewpoint, Trieste, 1990, pp. 373–463. World Scientific, Singapore (1991)

    Google Scholar 

  18. Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, Berlin (1982)

    Google Scholar 

  19. Brown, K.S.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Brown, K.S.: Buildings. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  21. Borel, A., Tits, J.: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27, 55–150 (1965)

    Article  MathSciNet  Google Scholar 

  22. Bruhat, F., Tits, J.: BN-paires de type affine et données radicielles. C. R. Acad. Sci. Paris Sér. A 263, 598–601 (1966)

    MATH  MathSciNet  Google Scholar 

  23. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local: I. Donées radicielles valuées. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  24. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local: II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math. 60, 5–184 (1984)

    Article  Google Scholar 

  25. Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. Fr. 112, 259–301 (1984)

    MATH  MathSciNet  Google Scholar 

  26. Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local, II. Groupes unitaires. Bull. Soc. Math. Fr. 115, 141–195 (1987)

    MATH  MathSciNet  Google Scholar 

  27. Cassels, J.: Local Fields. London Mathematical Society Student Texts, vol. 3. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  28. Chevalley, C.: Sur certains groupes simples. Tôhoku Math. J. (2) 7, 14–66 (1955)

    Google Scholar 

  29. Coxeter, H.S.M.: Discrete groups generated by reflections. Ann. Math. (2) 3, 588–621 (1933)

    Google Scholar 

  30. Davis, M.W.: The Geometry and Topology of Coxeter Groups. London Mathematical Society Monographs, vol. 32. Princeton University Press, Princeton (2008)

    Google Scholar 

  31. Dieudonné, J.: La Géométrie des Groupes Classiques. Springer, New York (1963)

    MATH  Google Scholar 

  32. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1994)

    Google Scholar 

  33. Grove, L., Benson, C.: Finite Reflecion Groups. Graduate Texts in Mathematics, vol. 99. Springer, Berlin (1985)

    Google Scholar 

  34. Geoghegan, R.: Topological Methods in Group Theory. Graduate Texts in Mathematics, vol. 243. Springer, Berlin (2008)

    Google Scholar 

  35. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  36. Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, Berlin (1981)

    Google Scholar 

  37. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  38. Iwahori, N., Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of \(\mathfrak{p}\)-adic Chevalley groups. Inst. Hautes Études Sci. Publ. Math. 25, 5–48 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  39. Jacobson, N.: Basic Algebra, I. W.H. Freeman and Co., San Francisco (1974)

    MATH  Google Scholar 

  40. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  41. Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Publ. Math. Inst. Hautes Études Sci. 86, 115–197 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  42. Lang, S.: Algebraic groups over finite fields. Am. J. Math. 78, 555–563 (1956)

    Article  MATH  Google Scholar 

  43. Lang, S.: Algebra. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  44. Mühlherr, B., Ronan, M.: Local to global structure in twin buildings. Invent. Math. 122, 71–81 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  45. Papadopoulos, A.: Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics, vol. 6. European Mathematical Society (EMS) (2005)

    Google Scholar 

  46. Platonov, V., Rapinchuk, A.: Algebraic Groups and Number Theory. Pure and Applied Mathematics, vol. 139. Academic, New York (1994)

    Google Scholar 

  47. Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs. Oxford University Press, Oxford (1986)

    MATH  Google Scholar 

  48. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149. Springer, Heidelberg (1994)

    Google Scholar 

  49. Rémy, B.: Groupes de Kac-Moody Déployés et Presque Déployés. Astérisque, vol. 277. Société Mathématique de France (2002)

    Google Scholar 

  50. Ronan, M.A.: Lectures on Buildings. Perspectives in Mathematics, vol. 7. Academic, New York (1989)

    Google Scholar 

  51. Rousseau, G.: Immeubles des groupes réducitifs sur les corps locaux. Ph.D. thesis, U.E.R. Mathématique, Université Paris XI, Orsay (1977)

    Google Scholar 

  52. Ronan, M.A., Tits, J.: Twin trees, I. Invent. Math. 116, 463–479 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  53. Ronan, M.A., Tits, J.: Twin trees, II. Local structure and a universal construction. Isr. J. Math. 109, 349–377 (1999)

    MATH  MathSciNet  Google Scholar 

  54. Serre, J.P.: Local Fields. Graduate Texts in Mathematics, vol. 67. Springer, Berlin (1979)

    Google Scholar 

  55. Spanier, E.H.: Algebraic Topology. Springer, Berlin (1966)

    MATH  Google Scholar 

  56. Springer, T.A.: Reductive groups. In: Automorphic Forms, Representations and L-Functions. Proceedings of the Symposium on Pure Mathematics, Oregon State University, Corvallis, 1977, Part 1, vol. XXXIII, pp. 3–27. American Mathematical Society, Providence (1979)

    Google Scholar 

  57. Springer, T.A.: Linear Algebraic Groups, 2nd edn. Progress in Mathematics, vol. 9. Birkhäuser, Boston (1998)

    Google Scholar 

  58. Seifert, H., Threlfall, W.: A Textbook of Topology. Academic, New York (1980)

    MATH  Google Scholar 

  59. Tits, J.: Sur certaines classes d’espaces homogènes de groupes de Lie. Acad. Roy. Belgique, Cl. Sci., Mém., Coll. 8∘ 29(3), 5–268 (1955)

    Google Scholar 

  60. Tits, J.: Les groupes de Lie exceptionnels et leur interpretation geométrique. Bull. Soc. Math. Belg. 8, 48–81 (1956)

    MATH  MathSciNet  Google Scholar 

  61. Tits, J.: Sur les analogues algébriques des groupes semi-simples complexes. Centre Belge Rech. Math., Colloque d’Algèbre supérieure, Bruxelles du 19 au 22 déc., 1956 (1957)

    Google Scholar 

  62. Tits, J.: Groupes semi-simples complexes et géométrie projective. Semin. Bourbaki 7 (1954/1955), No. 112 (1959)

    Google Scholar 

  63. Tits, J.: Groupes algébriques semi-simples et geometries associees. In: Algebraical and Topological Foundations of Geometry. Proceedings of the Colloquium, Utrecht, August 1959, pp. 175–192 (1962)

    MathSciNet  Google Scholar 

  64. Tits, J.: Théoreme de Bruhat et sous-groupes paraboliques. C. R. Acad. Sci. Paris 254, 2910–2912 (1962)

    MATH  MathSciNet  Google Scholar 

  65. Tits, J.: Géométries polyédriques et groupes simples. Atti della II Riunione del Groupement de Mathematiciens d’Expression Latine, Firenze-Bologna 1961, pp. 66–88 (1963)

    Google Scholar 

  66. Tits, J.: Algebraic and abstract simple groups. Ann. Math. (2) 80, 313–329 (1964)

    Google Scholar 

  67. Tits, J.: Classification of algebraic semisimple groups. In: Algebraic Groups and Discontinuous Subgroups. Proceedings of the Symposium on Pure Mathematics, Boulder, 1965, pp. 33–62. American Mathematical Society, Providence (1966)

    Google Scholar 

  68. Tits, J.: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol. 386. Springer, Berlin (1974)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  70. Tits, J.: Twin buildings and groups of Kac-Moody type. In: Groups, Combinatorics & Geometry (Durham, 1990). London Mathematical Society Lecture Note Series, vol. 165, pp. 249–286. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  71. Tits, J.: Œuvres. Collected Works, vols. I–IV. European Mathematical Society (EMS) (2013)

    Google Scholar 

  72. Tits, J., Weiss, R.: Moufang Polygons. Springer, New York (2002)

    Book  MATH  Google Scholar 

  73. van der Waerden, B.: Algebra, vol. II. Springer, New York (1991)

    Book  Google Scholar 

  74. Wall, C.T.C.: Finiteness conditions for CW-complexes. Ann. Math. (2) 81, 56–69 (1965)

    Google Scholar 

  75. Wall, C.T.C.: Finiteness conditions for CW complexes. II. Proc. R. Soc. Ser. A 295, 129–139 (1966)

    Article  Google Scholar 

  76. Weil, A.: Basic Number Theory. Die Grundlehren der Mathematischen Wissenschaften, vol. 144. Springer, Berlin (1974)

    Google Scholar 

  77. Weiss, R.: The Structure of Spherical Buildings. Princeton University Press, Princeton (2004)

    MATH  Google Scholar 

  78. Weiss, R.: The Structure of Affine Buildings. Annals of Mathematics Studies, vol. 168. Princeton University Press, Princeton (2009)

    Google Scholar 

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  1. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Münster, Nordrhein-Westfalen, Germany

    Stefan Witzel

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Witzel, S. (2014). Basic Definitions and Properties. In: Finiteness Properties of Arithmetic Groups Acting on Twin Buildings. Lecture Notes in Mathematics, vol 2109. Springer, Cham. https://doi.org/10.1007/978-3-319-06477-2_1

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