메뉴 건너뛰기




Volumn 49, Issue 1, 2012, Pages 113-162

Expander graphs in pure and applied mathematics

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84855763533     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-2011-01359-3     Document Type: Article
Times cited : (176)

References (187)
  • 2
    • 84855766168 scopus 로고    scopus 로고
    • Rank gradient, cost of groups and the rank versus Heegaard genus problem
    • to appear
    • M. Abért and N. Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Euro. Math. Soc., to appear.
    • (2020) J. Euro. Math. Soc.
    • Abért, M.1    Nikolov, N.2
  • 3
    • 38249028223 scopus 로고
    • Explicit construction of linear sized tolerant networks
    • N. Alon and Fan R.K. Chung, Explicit construction of linear sized tolerant networks, Discrete Mathematics 72 (1988), 15-19.
    • (1988) Discrete Mathematics , vol.72 , pp. 15-19
    • Alon, N.1    Chung, F.R.K.2
  • 5
    • 0025535126 scopus 로고
    • 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), 857-865, IEEE Comput. Soc. Press, Los Alamitos, CA
    • L. Babai, G. Hetyei, W.M. Kantor, A. Lubotzky, and A. Seress, On the diameter of finite groups, 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 10), 857-865, IEEE Comput. Soc. Press, Los Alamitos, CA, 1990.
    • (1990) On the diameter of finite groups
    • Babai, L.1    Hetyei, G.2    Kantor, W.M.3    Lubotzky, A.4    Seress, A.5
  • 6
    • 85009806236 scopus 로고
    • Small-diameter Cayley graphs for finite simple groups
    • L. Babai,W.M. Kantor and A. Lubotzky, Small-diameter Cayley graphs for finite simple groups, European J. Combin. 10 (1989), no. 6, 507-522.
    • (1989) European J. Combin. , vol.10 , Issue.6 , pp. 507-522
    • Babai, L.1    Kantor, W.M.2    Lubotzky, A.3
  • 8
    • 57849147754 scopus 로고    scopus 로고
    • On amenability of group algebras
    • L. Bartholdi, On amenability of group algebras. I, Israel J. Math. 168 (2008), 153-165.
    • (2008) I, Israel J. Math. , vol.168 , pp. 153-165
    • Bartholdi, L.1
  • 10
    • 0033476287 scopus 로고    scopus 로고
    • On the full C*-algebras of arithmetic groups and the congruence subgroup problem
    • M.B. Bekka, On the full C*-algebras of arithmetic groups and the congruence subgroup problem, Forum Math. 11 (1999), no. 6, 705-715.
    • (1999) Forum Math , vol.11 , Issue.6 , pp. 705-715
    • Bekka, M.B.1
  • 12
    • 62849098128 scopus 로고    scopus 로고
    • Expanders and dimensional expansion
    • J. Bourgain, Expanders and dimensional expansion, C. R. Math. Acad. Sci. Paris 347 (2009), no. 7-8, 357-362.
    • (2009) C. R. Math. Acad. Sci. Paris , vol.347 , Issue.7-8 , pp. 357-362
    • Bourgain, J.1
  • 13
    • 84855803804 scopus 로고    scopus 로고
    • New developments in combinatorial number theory and applications, European Congress of Mathematics
    • Zurich
    • J. Bourgain, New developments in combinatorial number theory and applications, European Congress of Mathematics, 233-251, Eur. Math. Soc., Zurich, 2010.
    • (2010) Eur. Math. Soc. , pp. 233-251
    • Bourgain, J.1
  • 14
    • 79959877524 scopus 로고    scopus 로고
    • A proof of the positive density conjecture for integer Apollonian circle packings
    • J. Bourgain and E. Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings, J. Amer. Math. Soc. 24 (2011), 945-967.
    • (2011) J. Amer. Math. Soc. , vol.24 , pp. 945-967
    • Bourgain, J.1    Fuchs, E.2
  • 15
    • 34447341450 scopus 로고    scopus 로고
    • Invariant measures and stiffness for non-abelian groups of toral automorphisms
    • J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Invariant measures and stiffness for non-abelian groups of toral automorphisms, C. R.Math. Acad. Sci. Paris 344 (2007), no. 12, 737-742.
    • (2007) C. R. Math. Acad. Sci. Paris , vol.344 , Issue.12 , pp. 737-742
    • Bourgain, J.1    Furman, A.2    Lindenstrauss, E.3    Mozes, S.4
  • 16
    • 44449104463 scopus 로고    scopus 로고
    • Uniform expansion bounds for Cayley graphs of SL2(Fp)
    • (2)
    • J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL2(Fp), Ann. of Math. (2) 167 (2008), no. 2, 625-642.
    • (2008) Ann. of Math. , vol.167 , Issue.2 , pp. 625-642
    • Bourgain, J.1    Gamburd, A.2
  • 17
    • 36849017888 scopus 로고    scopus 로고
    • On the spectral gap for finitely-generated subgroups of SU(2)
    • J. Bourgain and A. Gamburd, On the spectral gap for finitely-generated subgroups of SU(2), Invent. Math. 171 (2008), no. 1, 83-121.
    • (2008) Invent. Math. , vol.171 , Issue.1 , pp. 83-121
    • Bourgain, J.1    Gamburd, A.2
  • 18
    • 56649109576 scopus 로고    scopus 로고
    • Expansion and random walks in SLd(Z/pnZ). I
    • J. Bourgain and A. Gamburd, Expansion and random walks in SLd(Z/pnZ). I, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 987-1011.
    • (2008) J. Eur. Math. Soc. (JEMS) , vol.10 , Issue.4 , pp. 987-1011
    • Bourgain, J.1    Gamburd, A.2
  • 19
    • 74249116408 scopus 로고    scopus 로고
    • Expansion and random walks in SLd(Z/pnZ). II, with an appendix by J. Bourgain
    • J. Bourgain and A. Gamburd, Expansion and random walks in SLd(Z/pnZ). II, with an appendix by J. Bourgain, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 5, 1057-1103.
    • (2009) J. Eur. Math. Soc. (JEMS) , vol.11 , Issue.5 , pp. 1057-1103
    • Bourgain, J.1    Gamburd, A.2
  • 21
    • 76549128679 scopus 로고    scopus 로고
    • Affine linear sieve, expanders, and sumproduct
    • J. Bourgain, A. Gamburd and P. Sarnak, Affine linear sieve, expanders, and sumproduct, Invent. Math. 179 (2010), no. 3, 559-644.
    • (2010) Invent. Math. , vol.179 , Issue.3 , pp. 559-644
    • Bourgain, J.1    Gamburd, A.2    Sarnak, P.3
  • 23
    • 2442427378 scopus 로고    scopus 로고
    • A sum-product estimate in finite fields, and applications
    • J. Bourgain, N. Katz and T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), no. 1, 27-57.
    • (2004) Geom. Funct. Anal. , vol.14 , Issue.1 , pp. 27-57
    • Bourgain, J.1    Katz, N.2    Tao, T.3
  • 24
    • 78649305136 scopus 로고    scopus 로고
    • On representations of integers in thin subgroups of SL2Z
    • J. Bourgain and A. Kontorovich, On representations of integers in thin subgroups of SL2Z, Geom. Funct. Anal. (GAFA) 20 (2010), 1144-1174.
    • (2010) Geom. Funct. Anal. (GAFA) , vol.20 , pp. 1144-1174
    • Bourgain, J.1    Kontorovich, A.2
  • 35
  • 36
    • 0003285996 scopus 로고
    • Geometry and spectra of compact Riemann surfaces
    • 106. Birkhäuser Boston, Inc., Boston, MA
    • Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+454 pp.
    • (1992) Progress in Mathematics
    • Buser, P.1
  • 38
    • 0000397257 scopus 로고
    • On the representation of a larger even integer as the sum of a prime and the product of at most two primes
    • J.R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157-176.
    • (1973) Sci. Sinica , vol.16 , pp. 157-176
    • Chen, J.R.1
  • 39
    • 0037933477 scopus 로고    scopus 로고
    • Démonstration de la conjecture t
    • L. Clozel, Démonstration de la conjecture t, Invent. Math. 151 (2003), no. 2, 297-328.
    • (2003) Invent. Math. , vol.151 , Issue.2 , pp. 297-328
    • Clozel, L.1
  • 40
    • 13244290415 scopus 로고    scopus 로고
    • Elementary number theory, group theory, and Ramanujan graphs
    • Cambridge University Press, Cambridge
    • G. Davidoff, P. Sarnak and A. Valette, Elementary number theory, group theory, and Ramanujan graphs, London Mathematical Society Student Texts, 55. Cambridge University Press, Cambridge, 2003. x+144 pp.
    • (2003) London Mathematical Society Student Texts , vol.55
    • Davidoff, G.1    Sarnak, P.2    Valette, A.3
  • 41
    • 0040676602 scopus 로고    scopus 로고
    • Walks on generating sets of groups
    • P. Diaconis and L. Saloff-Coste, Walks on generating sets of groups, Invent. Math. 134 (1998), no. 2, 251-299.
    • (1998) Invent. Math. , vol.134 , Issue.2 , pp. 251-299
    • Diaconis, P.1    Saloff-Coste, L.2
  • 42
    • 79953814858 scopus 로고    scopus 로고
    • Growth in SL2 over finite fields
    • O. Dinai, Growth in SL2 over finite fields, J. Group Theory, 14 (2011) 273-297.
    • (2011) J. Group Theory , vol.14 , pp. 273-297
    • Dinai, O.1
  • 43
    • 0000660463 scopus 로고
    • The probability of generating the symmetric group
    • J.D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199-205.
    • (1969) Math. Z. , vol.110 , pp. 199-205
    • Dixon, J.D.1
  • 44
    • 33750510901 scopus 로고    scopus 로고
    • Finite covers of random 3-manifolds
    • N.M. Dunfield and W.P. Thurston, Finite covers of random 3-manifolds, Invent. Math. 166 (2006), no. 3, 457-521.
    • (2006) Invent. Math. , vol.166 , Issue.3 , pp. 457-521
    • Dunfield, N.M.1    Thurston, W.P.2
  • 46
    • 0038349206 scopus 로고    scopus 로고
    • The amenability of affine algebras
    • G. Elek, The amenability of affine algebras, J. Algebra 264 (2003), no. 2, 469-478.
    • (2003) J. Algebra , vol.264 , Issue.2 , pp. 469-478
    • Elek, G.1
  • 49
    • 56349086442 scopus 로고    scopus 로고
    • Golod-Shafarevich groups with property (T) and Kac-Moody groups
    • M. Ershov, Golod-Shafarevich groups with property (T) and Kac-Moody groups, Duke Math. J. 145 (2008), no. 2, 309-339.
    • (2008) Duke Math. J. , vol.145 , Issue.2 , pp. 309-339
    • Ershov, M.1
  • 50
    • 71949092331 scopus 로고    scopus 로고
    • Property (T) for noncommutative universal lattices
    • M. Ershov and A. Jaikin-Zapirain, Property (T) for noncommutative universal lattices, Invent. Math. 179 (2010), no. 2, 303-347.
    • (2010) Invent. Math. , vol.179 , Issue.2 , pp. 303-347
    • Ershov, M.1    Jaikin-Zapirain, A.2
  • 52
    • 84858018676 scopus 로고    scopus 로고
    • Opera de cribro, American Mathematical Society Colloquium Publications
    • Providence, RI
    • J. Friedlander and H. Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 2010. xx+527 pp.
    • (2010) American Mathematical Society , vol.57
    • Friedlander, J.1    Iwaniec, H.2
  • 53
    • 67949093089 scopus 로고    scopus 로고
    • A proof of Alon's second eigenvalue conjecture and related problems
    • J. Friedman, A proof of Alon's second eigenvalue conjecture and related problems, Mem. Amer. Math. Soc. 195 (2008), no. 910, viii+100 pp.
    • (2008) Mem. Amer. Math. Soc. , vol.195 , Issue.910
    • Friedman, J.1
  • 54
    • 84855766159 scopus 로고    scopus 로고
    • Ph.D. Thesis, Princeton University
    • E. Fuchs, Ph.D. Thesis, Princeton University.
    • Fuchs, E.1
  • 56
    • 0034340697 scopus 로고    scopus 로고
    • Côut des relations d'équivalence et des groupes
    • D. Gaboriau, Cô ut des relations d'équivalence et des groupes, Invent. Math. 139 (2000), no. 1, 41-98.
    • (2000) Invent. Math. , vol.139 , Issue.1 , pp. 41-98
    • Gaboriau, D.1
  • 57
    • 84855786638 scopus 로고    scopus 로고
    • What is Cost? Notices Amer
    • D. Gaboriau, What is Cost? Notices Amer. Math. Soc. 57 (2010), 1295-1296.
    • (2010) Math. Soc. , vol.57 , pp. 1295-1296
    • Gaboriau, D.1
  • 58
    • 0142168845 scopus 로고    scopus 로고
    • On the spectral gap for infinite index "congruence" subgroups of SL2(Z)
    • A. Gamburd, On the spectral gap for infinite index "congruence" subgroups of SL2(Z), Israel J. Math. 127 (2002), 157-200.
    • (2002) Israel J. Math. , vol.127 , pp. 157-200
    • Gamburd, A.1
  • 59
    • 13244261319 scopus 로고    scopus 로고
    • Expander graphs, random matrices and quantum chaos
    • 109-140, Walter de Gruyter GmbH & Co. KG, Berlin
    • A. Gamburd, Expander graphs, random matrices and quantum chaos, Random walks and geometry, 109-140, Walter de Gruyter GmbH & Co. KG, Berlin, 2004.
    • (2004) Random walks and geometry
    • Gamburd, A.1
  • 61
  • 63
    • 0009324237 scopus 로고
    • Factorizations in local subgroups of finite groups
    • American Mathematical Society, Providence, RI
    • G. Glauberman, Factorizations in local subgroups of finite groups, Regional Conference Series in Mathematics, No. 33. American Mathematical Society, Providence, RI, 1977. ix+74 pp.
    • (1977) Regional Conference Series in Mathematics , Issue.33
    • Glauberman, G.1
  • 64
    • 42249083052 scopus 로고    scopus 로고
    • Quasirandom groups
    • W.T. Gowers, Quasirandom groups, Combin. Probab. Comput. 17 (2008), no. 3, 363-387.
    • (2008) Combin. Probab. Comput. , vol.17 , Issue.3 , pp. 363-387
    • Gowers, W.T.1
  • 67
    • 43449086031 scopus 로고    scopus 로고
    • The primes contain arbitrarily long arithmetic progressions
    • B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481-547.
    • (2008) Ann. of Math. (2) , vol.167 , Issue.2 , pp. 481-547
    • Green, B.1    Tao, T.2
  • 68
    • 77957719058 scopus 로고    scopus 로고
    • Linear equations in primes
    • B. Green and T. Tao, Linear equations in primes, Ann. of Math. (2) 171 (2010), no. 3, 1753-1850.
    • (2010) Ann. of Math. (2) , vol.171 , Issue.3 , pp. 1753-1850
    • Green, B.1    Tao, T.2
  • 69
    • 84855819497 scopus 로고    scopus 로고
    • An inverse theorem for the Gowers Us+1[N]-norm
    • B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers Us+1[N]-norm.
    • Green, B.1    Tao, T.2    Ziegler, T.3
  • 70
    • 0038237671 scopus 로고    scopus 로고
    • Random walk in random groups
    • M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73-146.
    • (2003) Geom. Funct. Anal. , vol.13 , Issue.1 , pp. 73-146
    • Gromov, M.1
  • 71
    • 78649402676 scopus 로고    scopus 로고
    • Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles
    • M. Gromov, Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles, Geom. Funct. Anal. 19 (2009), no. 3, 743-841.
    • (2009) Geom. Funct. Anal. , vol.19 , Issue.3 , pp. 743-841
    • Gromov, M.1
  • 72
    • 77955278240 scopus 로고    scopus 로고
    • Singularities, expanders and topology of maps. II. From combinatorics to topology via algebraic isoperimetry
    • M. Gromov, Singularities, expanders and topology of maps. II. From combinatorics to topology via algebraic isoperimetry, Geom. Funct. Anal. 20 (2010), 416-526.
    • (2010) Geom. Funct. Anal. , vol.20 , pp. 416-526
    • Gromov, M.1
  • 74
    • 84855790580 scopus 로고    scopus 로고
    • Linear representations of the automorphism group of a free group
    • F. Grunewald and A. Lubotzky, Linear representations of the automorphism group of a free group, Geom. Funct. Anal. 18 (2009), no. 5, 1564-1608.
    • (2009) Geom. Funct. Anal. , vol.18 , Issue.5 , pp. 1564-1608
    • Grunewald, F.1    Lubotzky, A.2
  • 75
    • 33646862107 scopus 로고
    • Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes
    • G.H. Hardy and J.E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1-70.
    • (1923) Acta Math , vol.44 , Issue.1 , pp. 1-70
    • Hardy, G.H.1    Littlewood, J.E.2
  • 76
    • 44449169647 scopus 로고    scopus 로고
    • Growth and generation in SL2(Z/pZ)
    • H.A. Helfgott, Growth and generation in SL2(Z/pZ), Ann. of Math. (2) 167 (2008), no. 2, 601-623.
    • (2008) Ann. of Math. (2) , vol.167 , Issue.2 , pp. 601-623
    • Helfgott, H.A.1
  • 77
    • 79952123630 scopus 로고    scopus 로고
    • Growth in SL3(Z/pZ)
    • H.A. Helfgott, Growth in SL3(Z/pZ), J. Eur. Math. Soc. 13 (2011), 761-851.
    • (2011) J. Eur. Math. Soc. , vol.13 , pp. 761-851
    • Helfgott, H.A.1
  • 78
    • 0036435284 scopus 로고    scopus 로고
    • Counterexamples to the Baum-Connes conjecture
    • N. Higson, V. Lafforgue and G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330-354.
    • (2002) Geom. Funct. Anal. , vol.12 , Issue.2 , pp. 330-354
    • Higson, N.1    Lafforgue, V.2    Skandalis, G.3
  • 81
    • 12444323815 scopus 로고    scopus 로고
    • Analytic Number Theory, American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence
    • H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp.
    • (2004) RI
    • Iwaniec, H.1    Kowalski, E.2
  • 82
    • 84855815414 scopus 로고    scopus 로고
    • Splitting fields of characteristic polynomials of random elements in arithmetic groups
    • to appear
    • F. Jouve, E. Kowalski and D. Zywina, Splitting fields of characteristic polynomials of random elements in arithmetic groups, Israel J. of Math., to appear.
    • Israel J. of Math.
    • Jouve, F.1    Kowalski, E.2    Zywina, D.3
  • 83
    • 0001941330 scopus 로고
    • The probability of generating a finite classical group
    • W.M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), no. 1, 67-87.
    • (1990) Geom. Dedicata , vol.36 , Issue.1 , pp. 67-87
    • Kantor, W.M.1    Lubotzky, A.2
  • 84
    • 80755140651 scopus 로고    scopus 로고
    • Invariable generation and the Chebotarev invariant of a finite group
    • W. M. Kantor, A. Lubotzky and A. Shalev, Invariable generation and the Chebotarev invariant of a finite group, J. of Algebra, 348 (2011) 302-314.
    • (2011) J. of Algebra , vol.348 , pp. 302-314
    • Kantor, W.M.1    Lubotzky, A.2    Shalev, A.3
  • 85
    • 0037673298 scopus 로고    scopus 로고
    • Generic-case complexity, decision problems in group theory, and random walks
    • I. Kapovich, A. Miasnikov, P. Schupp and V. Shpilrain, Generic-case complexity, decision problems in group theory, and random walks, J. Algebra 264 (2003), no. 2, 665-694.
    • (2003) J. Algebra , vol.264 , Issue.2 , pp. 665-694
    • Kapovich, I.1    Miasnikov, A.2    Schupp, P.3    Shpilrain, V.4
  • 86
    • 6944252026 scopus 로고    scopus 로고
    • Average-case complexity and decision problems in group theory
    • I. Kapovich, A. Miasnikov, P. Schupp and V. Shpilrain, Average-case complexity and decision problems in group theory, Adv. Math. 190 (2005), no. 2, 343-359.
    • (2005) Adv. Math. , vol.190 , Issue.2 , pp. 343-359
    • Kapovich, I.1    Miasnikov, A.2    Schupp, P.3    Shpilrain, V.4
  • 87
    • 80052746851 scopus 로고    scopus 로고
    • On group-theoretic models of randomness and genericity
    • I. Kapovich and P. Schupp, On group-theoretic models of randomness and genericity, Groups Geom. Dyn. 2 (2008), no. 3, 383-404.
    • (2008) Groups Geom. Dyn. , vol.2 , Issue.3 , pp. 383-404
    • Kapovich, I.1    Schupp, P.2
  • 88
    • 34848914478 scopus 로고    scopus 로고
    • Universal lattices and unbounded rank expanders
    • M. Kassabov, Universal lattices and unbounded rank expanders, Invent. Math. 170 (2007), no. 2, 297-326.
    • (2007) Invent. Math. , vol.170 , Issue.2 , pp. 297-326
    • Kassabov, M.1
  • 89
    • 34848824034 scopus 로고    scopus 로고
    • Symmetric groups and expander graphs
    • M. Kassabov, Symmetric groups and expander graphs, Invent. Math. 170 (2007), no. 2, 327-354.
    • (2007) Invent. Math. , vol.170 , Issue.2 , pp. 327-354
    • Kassabov, M.1
  • 91
    • 33744540790 scopus 로고    scopus 로고
    • Universal lattices and property tau
    • M. Kassabov and N. Nikolov, Universal lattices and property tau, Invent. Math. 165 (2006), no. 1, 209-224.
    • (2006) Invent. Math. , vol.165 , Issue.1 , pp. 209-224
    • Kassabov, M.1    Nikolov, N.2
  • 94
    • 34250526463 scopus 로고
    • On the connection of the dual space of a group with the structure of its closed subgroups, (Russian) Funkcional
    • D.A. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, (Russian) Funkcional. Anal. i Priložen. 1 (1967), 71-74.
    • (1967) Anal. i Priložen. , vol.1 , pp. 71-74
    • Kazhdan, D.A.1
  • 95
    • 0037248070 scopus 로고    scopus 로고
    • Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, with appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak
    • H.H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, with appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Amer. Math. Soc. 16 (2003), no. 1, 139-183.
    • (2003) J. Amer. Math. Soc. , vol.16 , Issue.1 , pp. 139-183
    • Kim, H.H.1
  • 96
    • 0012782376 scopus 로고
    • On the realization of nets in 3-dimensional space
    • See also SelectedWorks of A.N. Kolmogorov, Vol. 3, pp. 194-202 (and a remark on page 245), Kluwer Academic Publishers, 1993
    • A. N. Kolmogorov and Y.M. Barzdin, On the realization of nets in 3-dimensional space, Probl. Cybernet, 8, 261-268, 1967. See also SelectedWorks of A.N. Kolmogorov, Vol. 3, pp. 194-202 (and a remark on page 245), Kluwer Academic Publishers, 1993.
    • (1967) Probl. Cybernet , vol.8 , pp. 261-268
    • Kolmogorov, A.N.1    Barzdin, Y.M.2
  • 97
    • 79953735111 scopus 로고    scopus 로고
    • Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds
    • A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc. 24 (2011), 603-648.
    • (2011) J. Amer. Math. Soc. , vol.24 , pp. 603-648
    • Kontorovich, A.1    Oh, H.2
  • 100
    • 84855799632 scopus 로고
    • Seminar Bourbaki Exp. no. November
    • E. Kowalski, Sieve in expansion, Seminar Bourbaki Exp. no. 1028, November 2010.
    • (1028) Sieve in expansion , pp. 2010
    • Kowalski, E.1
  • 101
    • 18444401389 scopus 로고    scopus 로고
    • Expanders, rank and graphs of groups
    • M. Lackenby, Expanders, rank and graphs of groups, Israel J. Math. 146 (2005), 357- 370.
    • (2005) Israel J. Math. , vol.146 , pp. 357-370
    • Lackenby, M.1
  • 102
    • 17444379306 scopus 로고    scopus 로고
    • A characterisation of large finitely presented groups
    • M. Lackenby, A characterisation of large finitely presented groups, J. Algebra 287 (2005), no. 2, 458-473.
    • (2005) J. Algebra , vol.287 , Issue.2 , pp. 458-473
    • Lackenby, M.1
  • 103
    • 33645508714 scopus 로고    scopus 로고
    • Heegaard splittings, the virtually Haken conjecture and property (t)
    • M. Lackenby, Heegaard splittings, the virtually Haken conjecture and property (t), Invent. Math. 164 (2006), no. 2, 317-359.
    • (2006) Invent. Math. , vol.164 , Issue.2 , pp. 317-359
    • Lackenby, M.1
  • 104
    • 68349133559 scopus 로고    scopus 로고
    • Large groups, property (t) and the homology growth of subgroups
    • M. Lackenby, Large groups, property (t) and the homology growth of subgroups, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 3, 625-648.
    • (2009) Math. Proc. Cambridge Philos. Soc. , vol.146 , Issue.3 , pp. 625-648
    • Lackenby, M.1
  • 105
    • 74249100149 scopus 로고    scopus 로고
    • LERF and the Lubotzky-Sarnak conjecture
    • M. Lackenby, D.D. Long and A.W. Reid, LERF and the Lubotzky-Sarnak conjecture, Geom. Topol. 12 (2008), no. 4, 2047-2056.
    • (2008) Geom. Topol. , vol.12 , Issue.4 , pp. 2047-2056
    • Lackenby, M.1    Long, D.D.2    Reid, A.W.3
  • 106
    • 77951205456 scopus 로고    scopus 로고
    • Covering spaces of arithmetic 3-orbifolds
    • IMRN 2008, no. 12, Art. ID rnn036
    • M. Lackenby, D.D. Long and A.W. Reid, Covering spaces of arithmetic 3-orbifolds, Int. Math. Res. Not. IMRN 2008, no. 12, Art. ID rnn036, 38 pp.
    • (2008) Int. Math. Res. , pp. 38
    • Lackenby, M.1    Long, D.D.2    Reid, A.W.3
  • 107
    • 84976040778 scopus 로고
    • On the cost of generating an equivalence relation
    • G. Levitt, On the cost of generating an equivalence relation, Ergodic Theory Dynam. Systems, 15(6) (1995), 1173-1181.
    • (1995) Ergodic Theory Dynam. Systems , vol.15 , Issue.6 , pp. 1173-1181
    • Levitt, G.1
  • 108
    • 0001056702 scopus 로고
    • The probability of generating a finite simple group
    • M.W. Liebeck and A. Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), no. 1, 103-113.
    • (1995) Geom. Dedicata , vol.56 , Issue.1 , pp. 103-113
    • Liebeck, M.W.1    Shalev, A.2
  • 110
    • 0344077919 scopus 로고    scopus 로고
    • Finite metric-spaces: combinatorics, geometry and algorithms
    • (Beijing, 2002), 573-586, Higher Ed. Press, Beijing
    • N. Linial, Finite metric-spaces: combinatorics, geometry and algorithms, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 573-586, Higher Ed. Press, Beijing, 2002.
    • (2002) Proceedings of the International Congress of Mathematicians , vol.3
    • Linial, N.1
  • 111
    • 44449147282 scopus 로고    scopus 로고
    • Heegaard genus and property t for hyperbolic 3-manifolds
    • D.D. Long, A. Lubotzky and A.W. Reid, Heegaard genus and property t for hyperbolic 3-manifolds, J. Topol. 1 (2008), no. 1, 152-158.
    • (2008) J. Topol. , vol.1 , Issue.1 , pp. 152-158
    • Long, D.D.1    Lubotzky, A.2    Reid, A.W.3
  • 112
    • 0042734182 scopus 로고    scopus 로고
    • Prym representations of mapping class groups
    • E. Looijenga, Prym representations of mapping class groups, Geom. Dedicata 64 (1997), no. 1, 69-83.
    • (1997) Geom. Dedicata , vol.64 , Issue.1 , pp. 69-83
    • Looijenga, E.1
  • 113
    • 0003421887 scopus 로고    scopus 로고
    • with an appendix by Jonathan D. Rogawski. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel
    • A. Lubotzky, Discrete groups, expanding graphs and invariant measures, with an appendix by Jonathan D. Rogawski. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2010. iii+192 pp.
    • (2010) Discrete groups, expanding graphs and invariant measures
    • Lubotzky, A.1
  • 114
    • 0002093056 scopus 로고
    • Surveys in Combinatorics, 1995 (Stirling), 155-189, London Math. Soc. Lecture Note Ser. 218. Cambridge Univ. Press, Cambridge
    • A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, Surveys in Combinatorics, 1995 (Stirling), 155-189, London Math. Soc. Lecture Note Ser., 218, Cambridge Univ. Press, Cambridge, 1995.
    • (1995) Cayley graphs: eigenvalues, expanders and random walks
    • Lubotzky, A.1
  • 115
    • 0030496386 scopus 로고    scopus 로고
    • Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem
    • A. Lubotzky, Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem, Ann. of Math. (2) 144 (1996), no. 2, 441-452.
    • (1996) Ann. of Math. (2) , vol.144 , Issue.2 , pp. 441-452
    • Lubotzky, A.1
  • 116
    • 0002892831 scopus 로고    scopus 로고
    • Free quotients and the first Betti number of some hyperbolic manifolds
    • A. Lubotzky, Free quotients and the first Betti number of some hyperbolic manifolds, Transform. Groups 1 (1996), no. 1-2, 71-82.
    • (1996) Transform. Groups , vol.1 , Issue.1-2 , pp. 71-82
    • Lubotzky, A.1
  • 117
    • 23444439212 scopus 로고    scopus 로고
    • What is... property (τ)
    • A. Lubotzky, What is... property (τ), Notices Amer. Math. Soc. 52 (2005), no. 6, 626-627.
    • (2005) Notices Amer. Math. Soc. , vol.52 , Issue.6 , pp. 626-627
    • Lubotzky, A.1
  • 118
    • 79960499659 scopus 로고    scopus 로고
    • Finite simple groups of Lie type as expanders
    • A. Lubotzky, Finite simple groups of Lie type as expanders, J. Eur. Math. Soc. 13 (2011), 1331-1341.
    • (2011) J. Eur. Math. Soc. , vol.13 , pp. 1331-1341
    • Lubotzky, A.1
  • 119
    • 84855799634 scopus 로고    scopus 로고
    • Sieve methods in group theory: I. powers in linear groups
    • to appear
    • A. Lubotzky and C. Meiri, Sieve methods in group theory: I. powers in linear groups. Geom. Dedicata, to appear.
    • Geom. Dedicata
    • Lubotzky, A.1    Meiri, C.2
  • 122
    • 0035581584 scopus 로고    scopus 로고
    • The product replacement algorithm and Kazhdan's property (T)
    • A. Lubotzky and I. Pak, The product replacement algorithm and Kazhdan's property (T), J. Amer. Math. Soc. 14 (2001), no. 2, 347-363.
    • (2001) J. Amer. Math. Soc. , vol.14 , Issue.2 , pp. 347-363
    • Lubotzky, A.1    Pak, I.2
  • 123
    • 0002284975 scopus 로고
    • Ramanujan conjecture and explicit construction of expanders
    • A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan conjecture and explicit construction of expanders, Proc. STOC. 86 (1986), 240-246.
    • (1986) Proc. STOC. , vol.86 , pp. 240-246
    • Lubotzky, A.1    Phillips, R.2    Sarnak, P.3
  • 125
    • 84990604340 scopus 로고
    • Hecke operators and distributing points on the sphere. I, Frontiers of the mathematical sciences: 1985 (New York, 1985)
    • A. Lubotzky, R. Phillips and P. Sarnak, Hecke operators and distributing points on the sphere. I, Frontiers of the mathematical sciences: 1985 (New York, 1985). Comm. Pure Appl. Math. 39 (1986), no. S, suppl., 149-186.
    • (1986) Comm. Pure Appl. Math. , vol.39 , Issue.SUPPL. , pp. 149-186
    • Lubotzky, A.1    Phillips, R.2    Sarnak, P.3
  • 128
    • 32544454962 scopus 로고    scopus 로고
    • Ramanujan complexes of type A{ogonek}d, Probability in Mathematics
    • A. Lubotzky, B. Samuels and U. Vishne, Ramanujan complexes of type A{ogonek}d, Probability in Mathematics. Israel J. Math. 149 (2005), 267-299.
    • (2005) Israel J. Math. , vol.149 , pp. 267-299
    • Lubotzky, A.1    Samuels, B.2    Vishne, U.3
  • 129
    • 14644412918 scopus 로고    scopus 로고
    • Explicit constructions of Ramanujan complexes of type A{ogonek}d
    • A. Lubotzky, B. Samuels and U. Vishne, Explicit constructions of Ramanujan complexes of type A{ogonek}d, European J. Combin. 26 (2005), no. 6, 965-993.
    • (2005) European J. Combin. , vol.26 , Issue.6 , pp. 965-993
    • Lubotzky, A.1    Samuels, B.2    Vishne, U.3
  • 130
    • 0037842329 scopus 로고    scopus 로고
    • Subgroup growth
    • Birkhäuser Verlag, Basel
    • A. Lubotzky and D. Segal, Subgroup growth, Progress in Mathematics, 212. Birkhäuser Verlag, Basel, 2003. xxii+453 pp.
    • (2003) Progress in Mathematics , vol.212
    • Lubotzky, A.1    Segal, D.2
  • 131
    • 84855787720 scopus 로고    scopus 로고
    • Finite representations in the unitary dual and Ramanujan groups, Discrete geometric analysis
    • Amer. Math. Soc. Providence, RI
    • A. Lubotzky and Y. Shalom, Finite representations in the unitary dual and Ramanujan groups, Discrete geometric analysis, 173-189, Contemp. Math., 347, Amer. Math. Soc., Providence, RI, 2004.
    • (2004) Contemp. Math. , vol.347 , pp. 173-189
    • Lubotzky, A.1    Shalom, Y.2
  • 133
    • 37149027788 scopus 로고    scopus 로고
    • Dimension expanders
    • A. Lubotzky and E. Zelmanov, Dimension expanders, J. Algebra 319 (2008), no. 2, 730-738.
    • (2008) J. Algebra , vol.319 , Issue.2 , pp. 730-738
    • Lubotzky, A.1    Zelmanov, E.2
  • 134
    • 51249171329 scopus 로고
    • Variants of Kazhdan's property for subgroups of semisimple groups
    • A. Lubotzky and R.J. Zimmer, Variants of Kazhdan's property for subgroups of semisimple groups, Israel J. Math. 66 (1989), no. 1-3, 289-299.
    • (1989) Israel J. Math. , vol.66 , Issue.1-3 , pp. 289-299
    • Lubotzky, A.1    Zimmer, R.J.2
  • 136
    • 79951561483 scopus 로고    scopus 로고
    • Random walks on the mapping class group
    • J. Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), 429-468.
    • (2011) Duke Math. J. , vol.156 , pp. 429-468
    • Maher, J.1
  • 137
    • 84055192621 scopus 로고    scopus 로고
    • Random Heegaard splittings
    • J. Maher, Random Heegaard splittings, J. Topology 3 (2010), 997-1025.
    • (2010) J. Topology , vol.3 , pp. 997-1025
    • Maher, J.1
  • 140
    • 51249182252 scopus 로고
    • Explicit constructions of graphs without short cycles and low density codes
    • G.A. Margulis, Explicit constructions of graphs without short cycles and low density codes, Combinatorica 2 (1982), no. 1, 71-78.
    • (1982) Combinatorica , vol.2 , Issue.1 , pp. 71-78
    • Margulis, G.A.1
  • 141
    • 0024045909 scopus 로고
    • Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators
    • G.A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problems of Information Transmission, 24(1):39-46, 1988.
    • (1988) Problems of Information Transmission , vol.24 , Issue.1 , pp. 39-46
    • Margulis, G.A.1
  • 143
    • 84855766160 scopus 로고    scopus 로고
    • A lecture given at Diaconis' 60th birthday. Available at
    • B. Mazur, It is a story, A lecture given at Diaconis' 60th birthday. Available at http://www.math.ucsd.edu/williams/diaconis/It.is.a.story.3.pdf
    • It is a story
    • Mazur, B.1
  • 144
    • 16244423667 scopus 로고    scopus 로고
    • Expanders in group algebras
    • R. Meshulam and A. Wigderson, Expanders in group algebras, Combinatorica 24 (2004), no. 4, 659-680.
    • (2004) Combinatorica , vol.24 , Issue.4 , pp. 659-680
    • Meshulam, R.1    Wigderson, A.2
  • 145
    • 0002622843 scopus 로고
    • Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q
    • M. Morgenstern, Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q, J. Combin. Theory Ser. B 62 (1994), 44-62.
    • (1994) J. Combin. Theory Ser. B , vol.62 , pp. 44-62
    • Morgenstern, M.1
  • 146
    • 84984549325 scopus 로고    scopus 로고
    • Generic complexity of undecidable problems
    • A.G. Myasnikov and A.N. Rybalov, Generic complexity of undecidable problems, J. Symbolic Logic 73 (2008), no. 2, 656-673.
    • (2008) J. Symbolic Logic , vol.73 , Issue.2 , pp. 656-673
    • Myasnikov, A.G.1    Rybalov, A.N.2
  • 147
    • 78650234932 scopus 로고    scopus 로고
    • Prime and almost prime integral points on principal homogeneous spaces
    • A. Nevo and P. Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361-402.
    • (2010) Acta Math , vol.205 , pp. 361-402
    • Nevo, A.1    Sarnak, P.2
  • 148
    • 33846960778 scopus 로고    scopus 로고
    • A product of decomposition for the classical quasisimple groups
    • N. Nikolov, A product of decomposition for the classical quasisimple groups, J. Group Theory 10 (2007), no. 1, 43-53.
    • (2007) J. Group Theory , vol.10 , Issue.1 , pp. 43-53
    • Nikolov, N.1
  • 150
    • 0000100717 scopus 로고
    • On subgroups of GLn(Fp)
    • M.V. Nori, On subgroups of GLn(Fp), Invent. Math. 88 (1987), no. 2, 257-275.
    • (1987) Invent. Math. , vol.88 , Issue.2 , pp. 257-275
    • Nori, M.V.1
  • 151
    • 0012956253 scopus 로고    scopus 로고
    • Strong approximation for Zariski dense subgroups over arbitrary global fields
    • R. Pink, Strong approximation for Zariski dense subgroups over arbitrary global fields, Comment. Math. Helv. 75 (2000), no. 4, 608-643.
    • (2000) Comment. Math. Helv. , vol.75 , Issue.4 , pp. 608-643
    • Pink, R.1
  • 152
  • 155
    • 0036000527 scopus 로고    scopus 로고
    • Entropy waves, the zig-zag graph product, and new constant-degree expanders
    • O. Reingold, S. Vadhan and A. Wigderson, Entropy waves, the zig-zag graph product, and new constant-degree expanders, Ann. of Math. (2) 155 (2002), no. 1, 157-187.
    • (2002) Ann. of Math. (2) , vol.155 , Issue.1 , pp. 157-187
    • Reingold, O.1    Vadhan, S.2    Wigderson, A.3
  • 156
    • 42549120299 scopus 로고    scopus 로고
    • Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms
    • I. Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353-379.
    • (2008) Duke Math. J. , vol.142 , Issue.2 , pp. 353-379
    • Rivin, I.1
  • 157
    • 77957977860 scopus 로고    scopus 로고
    • Zarisky density and genericity
    • I. Rivin, Zarisky density and genericity, Int. Math. Res. Not. IMRN 19 (2010), 3649- 3657.
    • (2010) Int. Math. Res. Not. IMRN , vol.19 , pp. 3649-3657
    • Rivin, I.1
  • 158
    • 34848918176 scopus 로고    scopus 로고
    • Iterative construction of Cayley expander graphs
    • E. Rozenman, A. Shalev and A. Wigderson, Iterative construction of Cayley expander graphs, Theory Comput. 2 (2006), 91-120.
    • (2006) Theory Comput , vol.2 , pp. 91-120
    • Rozenman, E.1    Shalev, A.2    Wigderson, A.3
  • 161
    • 0003288907 scopus 로고
    • Some applications of modular forms
    • Cambridge University Press, Cambridge
    • P. Sarnak, Some applications of modular forms, Cambridge Tracts in Mathematics, 99. Cambridge University Press, Cambridge, 1990. x+111 pp.
    • (1990) Cambridge Tracts in Mathematics , vol.99
    • Sarnak, P.1
  • 162
    • 84967734554 scopus 로고
    • Selberg's eigenvalue conjecture
    • P. Sarnak, Selberg's eigenvalue conjecture, Notices Amer. Math. Soc. 42 (1995), no. 11, 1272-1277.
    • (1995) Notices Amer. Math. Soc. , vol.42 , Issue.11 , pp. 1272-1277
    • Sarnak, P.1
  • 163
    • 23444458006 scopus 로고    scopus 로고
    • What is. . . an expander?
    • P. Sarnak, What is. .. an expander? Notices Amer. Math. Soc. 51 (2004), no. 7, 762-763.
    • (2004) Notices Amer. Math. Soc. , vol.51 , Issue.7 , pp. 762-763
    • Sarnak, P.1
  • 164
    • 66249115310 scopus 로고    scopus 로고
    • Equidistribution and primes
    • physique mathématiques, mathématiques et société. II. Astérisque No. 322
    • P. Sarnak, Equidistribution and primes, Géométrie differentielle, physique mathématiques, mathématiques et société. II. Astérisque No. 322 (2008), 225-240.
    • (2008) Géométrie differentielle , pp. 225-240
    • Sarnak, P.1
  • 166
    • 84855815417 scopus 로고    scopus 로고
    • Equidistribution Primes. Available at
    • P. Sarnak, Equidistribution and Primes, (2007) PIMS Lecture. Available at http://www.math.princeton.edu/sarnak/
    • (2007) PIMS Lecture
    • Sarnak, P.1
  • 167
    • 84855766163 scopus 로고    scopus 로고
    • Primes and orbits. Available at
    • P. Sarnak, Primes and orbits, MAA Garden State lecture. Available at http://www.math.princeton.edu/sarnak/
    • MAA Garden State lecture
    • Sarnak, P.1
  • 168
    • 84855766165 scopus 로고    scopus 로고
    • Integral Apollonian Packings - MAA Lecture January. Available at
    • P. Sarnak, Integral Apollonian Packings - MAA Lecture January 2009. Available at http://www.math.princeton
    • (2009)
    • Sarnak, P.1
  • 169
    • 0000579840 scopus 로고
    • Sur certaines hypoth'eses concernant les nombres premiers
    • erratum 5 (1958) 259
    • A. Schinzel and W. Sierpiński, Sur certaines hypoth'eses concernant les nombres premiers, (French) Acta Arith. 4 (1958), 185-208; erratum 5 (1958) 259.
    • (1958) (French) Acta Arith , vol.4 , pp. 185-208
    • Schinzel, A.1    Sierpiński, W.2
  • 171
    • 0002739011 scopus 로고
    • On the estimation of Fourier coefficients of modular forms
    • Amer. Math. Soc., Providence, RI
    • A. Selberg, On the estimation of Fourier coefficients of modular forms, 1965 Proc. Sympos. Pure Math., Vol. VIII, pp. 1-15, Amer. Math. Soc., Providence, RI
    • (1965) Proc. Sympos. Pure Math. , vol.8 , pp. 1-15
    • Selberg, A.1
  • 173
    • 0040056171 scopus 로고    scopus 로고
    • Expanding graphs and invariant means
    • Y. Shalom, Expanding graphs and invariant means, Combinatorica 17 (1997), no. 4, 555-575.
    • (1997) Combinatorica , vol.17 , Issue.4 , pp. 555-575
    • Shalom, Y.1
  • 174
    • 0142214312 scopus 로고    scopus 로고
    • Expander graphs and amenable quotients
    • (Minneapolis, MN, 1996) IMA Vol. Math. Appl., Springer, New York
    • Y. Shalom, Expander graphs and amenable quotients, Emerging applications of number theory (Minneapolis, MN, 1996), 571-581, IMA Vol. Math. Appl., 109, Springer, New York, 1999.
    • (1999) Emerging applications of number theory , vol.109 , pp. 571-581
    • Shalom, Y.1
  • 176
    • 34848878618 scopus 로고    scopus 로고
    • The algebraization of Kazhdan's property (T)
    • Eur. Math. Soc., Zurich
    • Y. Shalom, The algebraization of Kazhdan's property (T), International Congress of Mathematicians. Vol. II, 1283-1310, Eur. Math. Soc., Zurich, 2006.
    • (2006) International Congress of Mathematicians , vol.2 , pp. 1283-1310
    • Shalom, Y.1
  • 177
  • 178
    • 33947248273 scopus 로고    scopus 로고
    • Additive combinatorics
    • Cambridge University Press, Cambridge
    • T. Tao and V. Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006. xviii+512 pp.
    • Cambridge Studies in Advanced Mathematics , vol.105 , Issue.2006
    • Tao, T.1    Vu, V.2
  • 179
    • 0000644892 scopus 로고
    • Free subgroups in linear groups
    • J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972) 250-270.
    • (1972) J. Algebra , vol.20 , pp. 250-270
    • Tits, J.1
  • 181
    • 33748528695 scopus 로고    scopus 로고
    • Graphes de Ramanujan et applications, (French) Ramanujan graphs and applications
    • Astérisque No. Exp. No. 829, 4, 247-27
    • A. Valette, Graphes de Ramanujan et applications, (French) Ramanujan graphs and applications, Séminaire Bourbaki, Vol. 1996/97. Astérisque No. 245 (1997), Exp. No. 829, 4, 247-276.
    • (1997) Séminaire Bourbaki , vol.1996-1997 , Issue.245
    • Valette, A.1
  • 183
    • 84855819502 scopus 로고    scopus 로고
    • Expansion in SLd(OK/I), I square-free
    • P.O. Varju, Expansion in SLd(OK/I), I square-free
    • Varju, P.O.1
  • 184
    • 0000447570 scopus 로고
    • Strong approximation for Zariski-dense subgroups of semisimple algebraic groups
    • B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), no. 2, 271-315.
    • (1984) Ann. of Math. (2) , vol.120 , Issue.2 , pp. 271-315
    • Weisfeiler, B.1
  • 185
    • 0040406571 scopus 로고
    • Small eigenvalues of automorphic Laplacians in spaces of cusp forms
    • translation in J. Math. Sciences 36, No. 1, 106-114
    • P. Zograf, Small eigenvalues of automorphic Laplacians in spaces of cusp forms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 134 (1994), 157-168; translation in J. Math. Sciences 36, No. 1, 106-114.
    • (1994) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) , vol.134 , pp. 157-168
    • Zograf, P.1
  • 186
    • 0345618613 scopus 로고
    • A spectral proof of Rademacher's conjecture for congruence subgroups of the modular group
    • P. Zograf, A spectral proof of Rademacher's conjecture for congruence subgroups of the modular group, J. Reine Angew. Math. 414 (1991), 113-116.
    • (1991) J. Reine Angew. Math. , vol.414 , pp. 113-116
    • Zograf, P.1
  • 187
    • 0042853323 scopus 로고    scopus 로고
    • Property (T) and Kazhdan constants for discrete groups
    • A. Zuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643-670.
    • (2003) Geom. Funct. Anal. , vol.13 , Issue.3 , pp. 643-670
    • Zuk, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.