Israel Journal of Mathematics

, Volume 149, Issue 1, pp 267–299 | Cite as

Ramanujan complexes of typeà d



We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the buildings of typeà d−1 associated with PGL d (F) whereF is a local field of positive characteristic.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AT]
    E. Artin and J. Tate,Class Field Theory, W. A. Benjamin, New York, 1967.Google Scholar
  2. [B1]
    C. M. Ballantine,Ramanujan type buildings, Canadian Journal of Mathematics52 (2000), 1121–1148.MATHMathSciNetGoogle Scholar
  3. [B2]
    C. M. Ballantine,A hypergraph with commuting partial Laplacians, Canadian Mathematical Bulletin44 (2001), 385–397.MATHMathSciNetGoogle Scholar
  4. [Bu]
    D. Bump,Automorphic Forms and Representations, Cambridge Studies in Advances Mathematics55, Cambridge University Press, 1998.Google Scholar
  5. [BLS1]
    M. Burger, J.-S. Li and P. Sarnak,Ramanujan duals and automorphic spectrum, unpublished, 1990.Google Scholar
  6. [BLS2]
    M. Burger, J.-S. Li and P. Sarnak,Ramanujan duals and automorphic spectrum, Bulletin of the American Mathematical Society26 (1992), 253–257.MATHMathSciNetGoogle Scholar
  7. [C]
    P. Cartier,Representations of p-adic groups: A survey, Proceedings of Symposia in Pure Mathematics33 (1979), 111–155.MathSciNetGoogle Scholar
  8. [Cw]
    D. I. Cartwright,Spherical harmonic analysis on buildings of type à n, Monatshefte für Mathematik133 (2001), 93–109.MATHCrossRefMathSciNetGoogle Scholar
  9. [CM]
    D. I. Cartwright and W. Młotkowski,Harmonic analysis for groups acting on triangle buildings, Journal of the Australian Mathematical Society (A)56 (1994), 345–383.MATHCrossRefGoogle Scholar
  10. [CS]
    D. I. Cartwright and T. Steger,Elementary symmetric polynomials in numbers of modulus 1, Canadian Journal of Mathematics54 (2002), 239–262.MATHMathSciNetGoogle Scholar
  11. [CSZ]
    D. I. Cartwright, P. Solé and A. Żuk,Ramanujan geometries of type à n, Discrete Mathematics269 (2003), 35–43.MATHCrossRefMathSciNetGoogle Scholar
  12. [Gr]
    Y. Greenberg,On the spectrum of graphs and their universal covering (Hebrew), Doctoral Dissertation, The Hebrew University of Jerusalem, 1995.Google Scholar
  13. [GZ]
    R. I. Grigorchuk and A. Żuk,On the asymptotic spectrum of random walks on infinite families of graphs, inRandom Walks and Discrete Potential Theory (Cortona, 1997), Symposia Mathematica XXXIX, Cambridge University Press, Cambridge, 1999, pp. 188–204.Google Scholar
  14. [HT]
    R. Harris and R. Taylor,The Geometry and Cohomology of Simple Shimura Varieties, Annals of Mathematics Studies151, Princeton University Press, 2001.Google Scholar
  15. [JL]
    B. W. Jordan and R. Livne,The Ramanujan property for regular cubical complexes, Duke Mathematical Journal105 (2000), 85–103.MATHCrossRefMathSciNetGoogle Scholar
  16. [Kn]
    A. W. Knapp,Representation Theory of Semisimple Groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1986.MATHGoogle Scholar
  17. [L]
    L. Lafforgue,Chtoucas de Drinfeld et correspondance de Langlands (French), Inventiones Mathematicae147 (2002), 1–241.MATHCrossRefMathSciNetGoogle Scholar
  18. [LRS]
    G. Laumon, M. Rapoport and U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Inventiones Mathematicae113 (1993), 217–338.MATHCrossRefMathSciNetGoogle Scholar
  19. [Li]
    W.-C. W. Li,Ramanujan hypergraphs, Geometric and Functional Analysis14 (2004), 380–399.MATHCrossRefMathSciNetGoogle Scholar
  20. [Lu1]
    A. Lubotzky,Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics125, Birkhäuser, Basel, 1994.MATHGoogle Scholar
  21. [Lu2]
    A. Lubotzky,Cayley graphs: eigenvalues, expanders and random walks, inSurveys in Combinatorics (Stirling), London Mathematical Society Lecture Notes Series218, Cambridge University Press, 1995, pp. 155–189.Google Scholar
  22. [LPS]
    A. Lubotzky, R. Philips and P. Sarnak,Ramanujan graphs, Combinatorica8 (1988), 261–277.MATHCrossRefMathSciNetGoogle Scholar
  23. [LSV]
    A. Lubotzky, B. Samuels and U. Vishne,Explicit constructions of Ramanujan complexes of type à d, European Journal of Combinatorics, to appear.Google Scholar
  24. [M]
    I. G. Macdonald,Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Mathematical Monographs, Oxford University Press, 1995.Google Scholar
  25. [Ma1]
    G. Margulis,Explicit group-theoretic construction of combinatoric schemes and their applications in the construction of expanders and concentrators (Russian), Problemy Peredachi Informatsii24(1) (1988), 51–60; Engl. transl.: Problems of Information Transmission24(1) (1988), 39–46.MathSciNetGoogle Scholar
  26. [Ma2]
    G. Margulis,Discrete Subgroups of Semisimple Lie Groups, Results in Mathematics and Related Areas (3),17, Springer-Verlag, Berlin, 1991.MATHGoogle Scholar
  27. [Mo]
    M. Morgenstern,Existence and explicit constructions of q + 1regular Ramanujan graphs for every prime power q, Journal of Combinatorial Theory, Series B62 (1994), 44–62.MATHCrossRefMathSciNetGoogle Scholar
  28. [MW]
    C. Mœglin and J.-L. Waldspurger,Le spectre résidual de GLn, Annales Scientifiques de l’École Normale Supérieure, 4e sèrie22 (1989), 605–674.MATHGoogle Scholar
  29. [P]
    G. K. Pedersen,Analysis Now, GTM118, Springer, New York, 1989.Google Scholar
  30. [PR]
    V. Platonov and A. Rapinchuk,Algebraic Groups and Number Theory, Pure and Applied Mathematics139, Academic Press, Boston, 1994.MATHGoogle Scholar
  31. [Pr]
    G. Prasad,Strong approximation for semi-simple groups over function fields, Annals of Mathematics105 (1977), 553–572.CrossRefMathSciNetGoogle Scholar
  32. [R]
    M. Rapoport,The mathematical work of the 2002 Fields medalists: The work of Laurent Lafforgue, Notices of the American Mathematical Society50 (2003), 212–214.MATHMathSciNetGoogle Scholar
  33. [Ro]
    J. Rogawski,Representations of GLn and division algebras over a p-adic field, Duke Mathematical Journal50 (1983), 161–196.MATHCrossRefMathSciNetGoogle Scholar
  34. [Sa]
    A. Sarveniazi,Ramunajan (n 1,n 2, …,n d−1)-regular hypergraphs based on Bruhat-Tits buildings of type à d−1, Scholar
  35. [Sc]
    W. Scharlau,Quadratic Forms, Queen’s Papers in Pure and Applied Mathematics22, Queen’s University, Kingston, Ontario, 1969.MATHGoogle Scholar
  36. [Se1]
    J.-P. Serre,Galois Cohomology, (translated from the 1964 French text), Springer, Berlin, 1996.Google Scholar
  37. [Se2]
    J.-P. Serre,Le problème des groupes de congruence pour SL2, Annals of Mathematics92 (1970), 489–527.CrossRefMathSciNetGoogle Scholar
  38. [Se3]
    J.-P. Serre,Trees, 2nd edition, Springer Monographs in Mathematics, Springer, Berlin, 2003.MATHGoogle Scholar
  39. [T]
    M. Tadić,An external approach to unitary representations, Bulletin of the American Mathematical Society28 (1993), 215–252.MATHGoogle Scholar
  40. [V]
    M. F. Vigneras,Correspondances entre representations automorphes de GL(2)Sur une extension quadratique de GSp(4)sur Q, conjecture locale de Langlands pour GSp(4), Contemporary Mathematics53 (1986), 463–527.MathSciNetGoogle Scholar
  41. [Z]
    A. V. Zelevinsky,Induced representations of reductive p-adic groups II: on irreducible representations of GLn, Annales Scientifiques de l’École Normale Supérieure13 (1980), 165–210.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2005

Authors and Affiliations

  1. 1.Department of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations