×

zbMATH — the first resource for mathematics

Ramanujan complexes of type \(\widetilde A_d\). (English) Zbl 1087.05036
Summary: We define and construct Ramanujan complexes. These are simplicial complexes which are higher-dimensional analogues of Ramanujan graphs (constructed in [A. Lubotzky, R. Phillips and P. Sarnak, Combinatorica 8, 261–277 (1988; Zbl 0661.05035)]). They are obtained as quotients of the buildings of type \(\widetilde A_{d-1}\) associated with PGL\(_d(F)\) where \(F\) is a local field of positive characteristic.

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E42 Groups with a \(BN\)-pair; buildings
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] E. Artin and J. Tate,Class Field Theory, W. A. Benjamin, New York, 1967.
[2] C. M. Ballantine,Ramanujan type buildings, Canadian Journal of Mathematics52 (2000), 1121–1148. · Zbl 0999.20019
[3] C. M. Ballantine,A hypergraph with commuting partial Laplacians, Canadian Mathematical Bulletin44 (2001), 385–397. · Zbl 1039.11023
[4] D. Bump,Automorphic Forms and Representations, Cambridge Studies in Advances Mathematics55, Cambridge University Press, 1998.
[5] M. Burger, J.-S. Li and P. Sarnak,Ramanujan duals and automorphic spectrum, unpublished, 1990. · Zbl 0762.22009
[6] M. Burger, J.-S. Li and P. Sarnak,Ramanujan duals and automorphic spectrum, Bulletin of the American Mathematical Society26 (1992), 253–257. · Zbl 0762.22009
[7] P. Cartier,Representations of p-adic groups: A survey, Proceedings of Symposia in Pure Mathematics33 (1979), 111–155. · Zbl 0421.22010
[8] D. I. Cartwright,Spherical harmonic analysis on buildings of type à n , Monatshefte für Mathematik133 (2001), 93–109. · Zbl 1008.51019
[9] 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. · Zbl 0808.51014
[10] D. I. Cartwright and T. Steger,Elementary symmetric polynomials in numbers of modulus 1, Canadian Journal of Mathematics54 (2002), 239–262. · Zbl 1006.05059
[11] D. I. Cartwright, P. Solé and A. \.Zuk,Ramanujan geometries of type à n , Discrete Mathematics269 (2003), 35–43. · Zbl 1021.05068
[12] Y. Greenberg,On the spectrum of graphs and their universal covering (Hebrew), Doctoral Dissertation, The Hebrew University of Jerusalem, 1995.
[13] R. I. Grigorchuk and A. \.Zuk,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.
[14] R. Harris and R. Taylor,The Geometry and Cohomology of Simple Shimura Varieties, Annals of Mathematics Studies151, Princeton University Press, 2001. · Zbl 1036.11027
[15] B. W. Jordan and R. Livne,The Ramanujan property for regular cubical complexes, Duke Mathematical Journal105 (2000), 85–103. · Zbl 1009.05096
[16] A. W. Knapp,Representation Theory of Semisimple Groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1986. · Zbl 0604.22001
[17] L. Lafforgue,Chtoucas de Drinfeld et correspondance de Langlands (French), Inventiones Mathematicae147 (2002), 1–241. · Zbl 1038.11075
[18] G. Laumon, M. Rapoport and U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Inventiones Mathematicae113 (1993), 217–338. · Zbl 0809.11032
[19] W.-C. W. Li,Ramanujan hypergraphs, Geometric and Functional Analysis14 (2004), 380–399. · Zbl 1084.05047
[20] A. Lubotzky,Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics125, Birkhäuser, Basel, 1994. · Zbl 0826.22012
[21] 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. · Zbl 0835.05033
[22] A. Lubotzky, R. Philips and P. Sarnak,Ramanujan graphs, Combinatorica8 (1988), 261–277. · Zbl 0661.05035
[23] A. Lubotzky, B. Samuels and U. Vishne,Explicit constructions of Ramanujan complexes of type à d , European Journal of Combinatorics, to appear. · Zbl 1135.05038
[24] I. G. Macdonald,Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Mathematical Monographs, Oxford University Press, 1995. · Zbl 0824.05059
[25] 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.
[26] G. Margulis,Discrete Subgroups of Semisimple Lie Groups, Results in Mathematics and Related Areas (3),17, Springer-Verlag, Berlin, 1991. · Zbl 0732.22008
[27] 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. · Zbl 0814.68098
[28] C. Moeglin and J.-L. Waldspurger,Le spectre résidual de GL n , Annales Scientifiques de l’École Normale Supérieure, 4e sèrie22 (1989), 605–674. · Zbl 0696.10023
[29] G. K. Pedersen,Analysis Now, GTM118, Springer, New York, 1989. · Zbl 0668.46002
[30] V. Platonov and A. Rapinchuk,Algebraic Groups and Number Theory, Pure and Applied Mathematics139, Academic Press, Boston, 1994. · Zbl 0841.20046
[31] G. Prasad,Strong approximation for semi-simple groups over function fields, Annals of Mathematics105 (1977), 553–572. · Zbl 0348.22006
[32] M. Rapoport,The mathematical work of the 2002 Fields medalists: The work of Laurent Lafforgue, Notices of the American Mathematical Society50 (2003), 212–214. · Zbl 1059.01012
[33] J. Rogawski,Representations of GL n and division algebras over a p-adic field, Duke Mathematical Journal50 (1983), 161–196. · Zbl 0523.22015
[34] A. Sarveniazi,Ramunajan (n 1,n 2, ...,n d)-regular hypergraphs based on Bruhat-Tits buildings of type à d, arxiv.org/math.NT/0401181. · Zbl 1068.05048
[35] W. Scharlau,Quadratic Forms, Queen’s Papers in Pure and Applied Mathematics22, Queen’s University, Kingston, Ontario, 1969.
[36] J.-P. Serre,Galois Cohomology, (translated from the 1964 French text), Springer, Berlin, 1996.
[37] J.-P. Serre,Le problème des groupes de congruence pour SL2, Annals of Mathematics92 (1970), 489–527. · Zbl 0239.20063
[38] J.-P. Serre,Trees, 2nd edition, Springer Monographs in Mathematics, Springer, Berlin, 2003.
[39] M. Tadić,An external approach to unitary representations, Bulletin of the American Mathematical Society28 (1993), 215–252. · Zbl 0799.22010
[40] 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.
[41] A. V. Zelevinsky,Induced representations of reductive p-adic groups II: on irreducible representations of GL n , Annales Scientifiques de l’École Normale Supérieure13 (1980), 165–210. · Zbl 0441.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.