Ramanujan complexes and high dimensional expanders

Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last four decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.

[1]  Y. Ihara.  On discrete subgroups of the two by two projective linear group over p-adic fields , 1966 .

[2]  P. Sarnak.  Some Applications of Modular Forms , 1990 .

[3]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[5]  Roy Meshulam,et al.  A Moore bound for simplicial complexes , 2007 .

[6]  R. Meshulam,et al.  Homological connectivity of random k-dimensional complexes , 2009, Random Struct. Algorithms.

[7]  W. Li,et al.  The zeta functions of complexes from PGL(3): A representation-theoretic approach , 2008, 0809.1401.

[8]  Roy Meshulam,et al.  Random Latin squares and 2-dimensional expanders , 2013, 1307.3582.

[9]  Y. Cho,et al.  Discrete Groups , 1994 .

[10]  Imre Bárány,et al.  A generalization of carathéodory's theorem , 1982, Discret. Math..

[11]  Toshikazu Sunada,et al.  Fundamental groups and Laplacians , 1988 .

[12]  THE IHARA–SELBERG ZETA FUNCTION FOR PGL3 AND HECKE OPERATORS , 2004, math/0407509.

[13]  Donald I. Cartwright,et al.  A family of Ãn-groups , 1998 .

[14]  Jirí Matousek,et al.  On Gromov’s Method of Selecting Heavily Covered Points , 2011, Discret. Comput. Geom..

[15]  M. Gromov.  Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[16]  Wen-Ching Winnie Li.  Ramanujan hypergraphs , 2004 .

[17]  A. Mohammadi,et al.  Discrete subgroups acting transitively on vertices of a Bruhat–Tits building , 2012 .

[18]  V. Drinfel'd,et al.  The proof of peterson's conjecture for GL(2) over a global field of characteristic p , 1988 .

[19]  Roman N. Karasev,et al.  A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem , 2010, Discret. Comput. Geom..

[20]  Tali Kaufman,et al.  Edge transitive ramanujan graphs and symmetric LDPC good codes , 2012, STOC '12.

[21]  Patrick Solé,et al.  Ramanujan geometries of type Ãn , 2003, Discret. Math..

[22]  Noga Alon,et al.  On the second eigenvalue of a graph , 1991, Discret. Math..

[23]  Christopher K. Storm.  The Zeta Function of a Hypergraph , 2006, Electron. J. Comb..

[24]  Yuri Rabinovich,et al.  On multiplicative λ-approximations and some geometric applications , 2012, SODA.

[25]  Moshe Morgenstern,et al.  Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q , 1994, J. Comb. Theory, Ser. B.

[26]  E. M. Elsayed,et al.  DIFFERENCE EQUATIONS , 2001 .

[27]  Harry Kesten,et al.  Symmetric random walks on groups , 1959 .

[28]  Ilan Newman,et al.  On Multiplicative Lambda-Approximations and Some Geometric Applications , 2013, SIAM J. Comput..

[29]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[30]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[31]  Alireza Sarveniazi,et al.  Explicit construction of a Ramanujan $(n_1,n_2,\ldots,n_{d-1})$-regular hypergraph , 2007 .

[32]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[33]  A. Zuk,et al.  La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres , 1996 .

[34]  János Pach.  A Tverberg-type result on multicolored simplices , 1998, Comput. Geom..

[35]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[36]  Uzi Vishne,et al.  Ramanujan Complexes of Type $\tilde{A_d}$ , 2004 .

[37]  János Pach,et al.  Overlap properties of geometric expanders , 2011, SODA '11.

[38]  M. Murty.  Ramanujan Graphs , 1965 .

[39]  Matthew Kahle,et al.  Coboundary expanders , 2010, 1012.5316.

[40]  Uli Wagner,et al.  Minors in random and expanding hypergraphs , 2011, SoCG '11.

[41]  I. James,et al.  Singularities * , 2008 .

[42]  A. Borel,et al.  Cohomologie de certains groupes discrets et laplacien $p$-adique , 1974 .

[43]  Alexander Lubotzky,et al.  Ramanujan complexes of typeÃd , 2005 .

[44]  Ming-Hsuan Kang,et al.  Zeta Functions of Complexes Arising from PGL(3) , 2008, 0804.2305.

[45]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[46]  Alexander Lubotzky,et al.  Expander graphs in pure and applied mathematics , 2011, 1105.2389.

[47]  R. M. Tanner.  Explicit Concentrators from Generalized N-Gons , 1984 .

[48]  Anna Gundert,et al.  On laplacians of random complexes , 2012, SoCG '12.

[49]  P. Cartier.  Representations of p-adic groups: a survey , 1977 .

[50]  Anna Gundert,et al.  Higher Dimensional Cheeger Inequalities , 2014, SoCG.

[51]  Alain Valette,et al.  Graphes de Ramanujan et applications , 1997 .

[52]  A. Lubotzky,et al.  RAMANUJAN COMPLEXES OF TYPE Ad , 2007 .

[53]  Cristina Ballantine,et al.  Ramanujan Type Buildings , 2000, Canadian Journal of Mathematics.

[54]  Moshe Morgenstern,et al.  Natural bounded concentrators , 1995, Comb..

[55]  R. Chapman.  SOME APPLICATIONS OF MODULAR FORMS (Cambridge Tracts in Mathematics 99) , 1992 .

[56]  K. Hashimoto.  Zeta functions of finite graphs and representations of p-adic groups , 1989 .

[57]  F. H. Jackson.  q-Difference Equations , 1910 .

[58]  Alexander Lubotzky,et al.  Mixing Properties and the Chromatic Number of Ramanujan Complexes , 2014, 1407.7700.

[59]  Alexander Lubotzky,et al.  Explicit constructions of Ramanujan complexes of type , 2005, Eur. J. Comb..

[60]  Uzi Vishne,et al.  Explicit Constructions of Ramanujan Complexes of Type $\tilde{A}_d$ , 2004 .

[61]  Caroline J. Klivans,et al.  A Cheeger-Type Inequality on Simplicial Complexes , 2012, Adv. Appl. Math..

[62]  Laurent Lafforgue,et al.  Chtoucas de Drinfeld et correspondance de Langlands , 2002 .

[63]  Z. Füredi,et al.  The number of triangles covering the center of an n-set , 1984 .

[64]  Roy Meshulam,et al.  Expansion of Building-Like Complexes , 2014 .

[65]  Tali Kaufman,et al.  High dimensional expanders and property testing , 2014, ITCS.

[66]  Moshe Morgenstern.  Ramanujan Diagrams , 1994, SIAM J. Discret. Math..

[67]  Konstantin Golubev,et al.  Spectrum and combinatorics of Ramanujan triangle complexes , 2014 .

[68]  Tali Kaufman,et al.  Ramanujan Complexes and Bounded Degree Topological Expanders , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[69]  Howard Garland,et al.  p-Adic Curvature and the Cohomology of Discrete Subgroups of p-Adic Groups , 1973 .

[70]  Nathan Linial,et al.  Homological Connectivity Of Random 2-Complexes , 2006, Comb..