Counting arithmetic lattices and surfaces

Abstract

We give estimates on the number $\operatorname{AL}_H(x)$ of conjugacy classes of arithmetic lattices $\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic $3$-manifolds of volume at most $x$. Our main result is for the classical case $H=\operatorname{PSL}(2,\mathbb{R})$ where we show that \[ \lim_{x\to\infty}\frac{\log \operatorname{AL}_H(x)}{x\log x}=\frac{1}{2\pi}. \] The proofs use several different techniques: geometric (bounding the number of generators of $\Gamma$ as a function of its covolume), number theoretic (bounding the number of maximal such $\Gamma$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $\Gamma$).

  • [Bal] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, Basel: Birkhäuser, 1995, vol. 25.
    @book {Bal, MRKEY = {1377265},
      AUTHOR = {Ballmann, Werner},
      TITLE = {Lectures on Spaces of Nonpositive Curvature},
      SERIES = {DMV Seminar},
      VOLUME= {25},
      NOTE = {With an appendix by Misha Brin},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1995},
      PAGES = {viii+112},
      ISBN = {3-7643-5242-6},
      MRCLASS = {53C21 (58F17)},
      MRNUMBER = {97a:53053},
      MRREVIEWER = {Boris Hasselblatt},
      ZBLNUMBER = {0834.53003},
      }
  • [BGS] W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of Nonpositive Curvature, Boston, MA: Birkhäuser, 1985, vol. 61.
    @book {BGS, MRKEY = {823981},
      AUTHOR = {Ballmann, Werner and Gromov, Mikhael and Schroeder, Viktor},
      TITLE = {Manifolds of Nonpositive Curvature},
      SERIES = {Progr. Math.},
      VOLUME = {61},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1985},
      PAGES = {vi+263},
      ISBN = {0-8176-3181-X},
      MRCLASS = {53C20},
      MRNUMBER = {87h:53050},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      ZBLNUMBER = {0591.53001},
      }
  • [Be] Go to document M. Belolipetsky, "Counting maximal arithmetic subgroups," Duke Math. J., vol. 140, iss. 1, pp. 1-33, 2007.
    @article {Be, MRKEY = {2355066},
      AUTHOR = {Belolipetsky, Mikhail},
      TITLE = {Counting maximal arithmetic subgroups},
      NOTE = {With an appendix by J. Ellenberg and A. Venkatesh},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {140},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {1--33},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {22E40 (20G20)},
      MRNUMBER = {2008i:22005},
      MRREVIEWER = {Bachir Bekka},
      DOI = {10.1215/S0012-7094-07-14011-0},
      ZBLNUMBER = {1131.22008},
      }
  • [BL] M. Belolipetsky and A. Lubotsky, Counting manifolds and class field towers.
    @misc{BL,
      author = {Belolipetsky, Mikhai and Lubotsky, A.},
      TITLE={Counting manifolds and class field towers},
      ARXIV={0905.1841v1},
      }
  • [Bo] Go to document A. Borel, "Commensurability classes and volumes of hyperbolic $3$-manifolds," Ann. Scuola Norm. Sup. Pisa Cl. Sci., vol. 8, iss. 1, pp. 1-33, 1981.
    @article {Bo, MRKEY = {616899},
      AUTHOR = {Borel, A.},
      TITLE = {Commensurability classes and volumes of hyperbolic {$3$}-manifolds},
      JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci.},
      FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV},
      VOLUME = {8},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {1--33},
      CODEN = {PSNAAI},
      MRCLASS = {22E40 (20G30 51M25 57N10)},
      MRNUMBER = {82j:22008},
      MRREVIEWER = {Avner Ash},
      URL = {http://www.numdam.org/item?id=ASNSP_1981_4_8_1_1_0},
      ZBLNUMBER = {0473.57003},
      }
  • [BP] Go to document A. Borel and G. Prasad, "Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups," Inst. Hautes Études Sci. Publ. Math., iss. 69, pp. 119-171, 1989.
    @article {BP, MRKEY = {1019963},
      AUTHOR = {Borel, Armand and Prasad, Gopal},
      TITLE = {Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {69},
      YEAR = {1989},
      PAGES = {119--171},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {22E40 (11E72 20G30)},
      MRNUMBER = {91c:22021},
      MRREVIEWER = {Fritz Grunewald},
      URL = {http://www.numdam.org/item?id=PMIHES_1989__69__119_0},
      ZBLNUMBER = {0707.11032},
      }
  • [BoTu] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, New York: Springer-Verlag, 1982, vol. 82.
    @book {BoTu, MRKEY = {658304},
      AUTHOR = {Bott, Raoul and Tu, Loring W.},
      TITLE = {Differential Forms in Algebraic Topology},
      SERIES = {Grad. Texts Math.},
      VOLUME = {82},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {xiv+331},
      ISBN = {0-387-90613-4},
      MRCLASS = {57R19 (55-02 58-01 58A12)},
      MRNUMBER = {83i:57016},
      MRREVIEWER = {Hansklaus Rummler},
      ZBLNUMBER = {0496.55001},
      }
  • [BGLM] Go to document M. Burger, T. Gelander, A. Lubotzky, and S. Mozes, "Counting hyperbolic manifolds," Geom. Funct. Anal., vol. 12, iss. 6, pp. 1161-1173, 2002.
    @article {BGLM, MRKEY = {1952926},
      AUTHOR = {Burger, M. and Gelander, T. and Lubotzky, A. and Mozes, S.},
      TITLE = {Counting hyperbolic manifolds},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {12},
      YEAR = {2002},
      NUMBER = {6},
      PAGES = {1161--1173},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {22E40 (57N16)},
      MRNUMBER = {2003j:22014},
      MRREVIEWER = {Igor Rivin},
      DOI = {10.1007/s00039-002-1161-1},
      ZBLNUMBER = {1029.57021},
      }
  • [CF] Go to document T. Chinburg and E. Friedman, "The smallest arithmetic hyperbolic three-orbifold," Invent. Math., vol. 86, iss. 3, pp. 507-527, 1986.
    @article {CF, MRKEY = {860679},
      AUTHOR = {Chinburg, Ted and Friedman, Eduardo},
      TITLE = {The smallest arithmetic hyperbolic three-orbifold},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {86},
      YEAR = {1986},
      NUMBER = {3},
      PAGES = {507--527},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E40 (57N10)},
      MRNUMBER = {88a:22022},
      MRREVIEWER = {O. V. Shvartsman},
      DOI = {10.1007/BF01389265},
      ZBLNUMBER = {0643.57011},
      }
  • [EV] Go to document J. S. Ellenberg and A. Venkatesh, "The number of extensions of a number field with fixed degree and bounded discriminant," Ann. of Math., vol. 163, iss. 2, pp. 723-741, 2006.
    @article {EV, MRKEY = {2199231},
      AUTHOR = {Ellenberg, Jordan S. and Venkatesh, Akshay},
      TITLE = {The number of extensions of a number field with fixed degree and bounded discriminant},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {723--741},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11R45 (11R29)},
      MRNUMBER = {2006j:11159},
      MRREVIEWER = {T. Mets{ä}nkyl{ä}},
      DOI = {10.4007/annals.2006.163.723},
      ZBLNUMBER = {1099.11068},
      }
  • [E] A. Eisenmann, "Counting arithmetic subgroups and subgroup growth of virtually free groups," PhD Thesis , Hebrew University, 2010.
    @phdthesis{E,
      author={Eisenmann, A.},
      TITLE={Counting arithmetic subgroups and subgroup growth of virtually free groups},
      YEAR={2010},
      SCHOOL={Hebrew University},
      }
  • [Fi-Gr] Go to document T. Finis, F. Grunewald, and P. Tirao, "The cohomology of lattices in ${ SL}(2,\Bbb C)$," Experiment. Math., vol. 19, iss. 1, pp. 29-63, 2010.
    @article {Fi-Gr, MRKEY = {2649984},
      AUTHOR = {Finis, Tobias and Grunewald, Fritz and Tirao, Paulo},
      TITLE = {The cohomology of lattices in {${\rm SL}(2,\Bbb C)$}},
      JOURNAL = {Experiment. Math.},
      FJOURNAL = {Experimental Mathematics},
      VOLUME = {19},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {29--63},
      ISSN = {1058-6458},
      MRCLASS = {11F75 (11F72 30F40)},
      MRNUMBER = {2649984},
      URL = {http://projecteuclid.org/getRecord?id=euclid.em/1268404802},
      ZBLNUMBER = {05689261},
      }
  • [FL] S. Fomin and N. Lulov, "On the number of rim hook tableaux," Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$, vol. 223, iss. Teor. Predstav. Din. Sistemy, Kombin. i Algoritm. Metody. I, pp. 219-226, 340, 1995.
    @article {FL, MRKEY = {1374321},
      AUTHOR = {Fomin, Sergey and Lulov, Nathan},
      TITLE = {On the number of rim hook tableaux},
      JOURNAL = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. MatematicheskiĭInstitut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
      VOLUME = {223},
      YEAR = {1995},
      NUMBER = {Teor. Predstav. Din. Sistemy, Kombin. i Algoritm. Metody. I},
      PAGES = {219--226, 340},
      ISSN = {0373-2703},
      MRCLASS = {05E10 (05A20 20C30)},
      MRNUMBER = {97g:05168},
      MRREVIEWER = {Witold Kra{\'s}kiewicz},
      ZBLNUMBER = {0909.05046},
      }
  • [Ge] Go to document T. Gelander, "Homotopy type and volume of locally symmetric manifolds," Duke Math. J., vol. 124, iss. 3, pp. 459-515, 2004.
    @article {Ge, MRKEY = {2084613},
      AUTHOR = {Gelander, Tsachik},
      TITLE = {Homotopy type and volume of locally symmetric manifolds},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {124},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {459--515},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {53C24 (22E40 53C20 57N16)},
      MRNUMBER = {2005i:53050},
      MRREVIEWER = {Dave Witte Morris},
      DOI = {10.1215/S0012-7094-04-12432-7},
      ZBLNUMBER = {1076.53040},
      }
  • [Ge:morse] T. Gelander, Volume vs rank of lattices.
    @misc{Ge:morse,
      author={Gelander, Tsachik},
      TITLE = {Volume vs rank of lattices},
      NOTE={preprint},
      }
  • [GLNP] Go to document D. Goldfeld, A. Lubotzky, N. Nikolov, and L. Pyber, "Counting primes, groups, and manifolds," Proc. Natl. Acad. Sci. USA, vol. 101, iss. 37, pp. 13428-13430, 2004.
    @article {GLNP, MRKEY = {2226643},
      AUTHOR = {Goldfeld, Dorian and Lubotzky, Alexander and Nikolov, Nikolay and Pyber, L{á}szl{ó}},
      TITLE = {Counting primes, groups, and manifolds},
      JOURNAL = {Proc. Natl. Acad. Sci. USA},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {101},
      YEAR = {2004},
      NUMBER = {37},
      PAGES = {13428--13430},
      ISSN = {1091-6490},
      CODEN = {PNASFB},
      MRCLASS = {11N45 (20E07 20H05 53C20)},
      MRNUMBER = {2007b:11144},
      MRREVIEWER = {Lucy Lifschitz},
      DOI = {10.1073/pnas.0404571101},
      ZBLNUMBER = {1139.11324},
      }
  • [Hu] A. Hurwitz, "Über die Anzahl der Riemannschen Flächen mit gegebener Verzweigungspunkten," Math. Ann., vol. 55, pp. 53-66, 1902.
    @article{Hu,
      author={Hurwitz, A.},
      TITLE={Über die Anzahl der Riemannschen Flächen mit gegebener Verzweigungspunkten},
      JOURNAL={Math. Ann.},
      VOLUME={55},
      YEAR={1902},
      PAGES={53--66},
      JFMNUMBER={32.0404.04},
      }
  • [KM] D. A. Kazhdan and G. A. Margulis, "A proof of Selberg’s hypothesis," Math. Sbornik, vol. 75, pp. 162-168, 1968.
    @article{KM,
      author={Kazhdan, D. A. and Margulis, G. A.},
      TITLE={A proof of Selberg's hypothesis},
      JOURNAL={Math. Sbornik},
      VOLUME={75},
      YEAR={1968},
      PAGES={162--168},
      NOTE={(in Russian)},
      MRNUMBER={36 \#6535},
      ZBLNUMBER={0241.22024},
      }
  • [LiSh] Go to document M. W. Liebeck and A. Shalev, "Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks," J. Algebra, vol. 276, iss. 2, pp. 552-601, 2004.
    @article {LiSh, MRKEY = {2058457},
      AUTHOR = {Liebeck, Martin W. and Shalev, Aner},
      TITLE = {Fuchsian groups, coverings of {R}iemann surfaces, subgroup growth, random quotients and random walks},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {276},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {552--601},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20H10 (20C30 20E07 20P05 28C10 57M07 60B15 60G50)},
      MRNUMBER = {2005e:20076},
      MRREVIEWER = {Goulnara N. Arzhantseva},
      DOI = {10.1016/S0021-8693(03)00515-5},
      ZBLNUMBER = {1068.20052},
      }
  • [Lub1] Go to document A. Lubotzky, "Subgroup growth and congruence subgroups," Invent. Math., vol. 119, iss. 2, pp. 267-295, 1995.
    @article {Lub1, MRKEY = {1312501},
      AUTHOR = {Lubotzky, Alexander},
      TITLE = {Subgroup growth and congruence subgroups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {119},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {267--295},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20H05 (20G30)},
      MRNUMBER = {95m:20054},
      MRREVIEWER = {H. Bass},
      DOI = {10.1007/BF01245183},
      ZBLNUMBER = {0848.20036},
      }
  • [Lub2] Go to document A. Lubotzky, "Free quotients and the first Betti number of some hyperbolic manifolds," Transform. Groups, vol. 1, iss. 1-2, pp. 71-82, 1996.
    @article {Lub2, MRKEY = {1390750},
      AUTHOR = {Lubotzky, Alexander},
      TITLE = {Free quotients and the first {B}etti number of some hyperbolic manifolds},
      JOURNAL = {Transform. Groups},
      FJOURNAL = {Transformation Groups},
      VOLUME = {1},
      YEAR = {1996},
      NUMBER = {1-2},
      PAGES = {71--82},
      ISSN = {1083-4362},
      MRCLASS = {57M60 (20E06 22E40 57M50)},
      MRNUMBER = {97d:57016},
      MRREVIEWER = {Igor Rivin},
      DOI = {10.1007/BF02587736},
      ZBLNUMBER = {0876.22015},
      }
  • [LS] A. Lubotzky and D. Segal, Subgroup Growth, Basel: Birkhäuser, 2003, vol. 212.
    @book {LS, MRKEY = {1978431},
      AUTHOR = {Lubotzky, Alexander and Segal, Dan},
      TITLE = {Subgroup Growth},
      SERIES = {Progr. Math.},
      VOLUME = {212},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2003},
      PAGES = {xxii+453},
      ISBN = {3-7643-6989-2},
      MRCLASS = {20E07 (20E18 20E26 20F69)},
      MRNUMBER = {2004k:20055},
      MRREVIEWER = {Avinoam Mann},
      ZBLNUMBER = {1071.20033},
      }
  • [MR] C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, New York: Springer-Verlag, 2003, vol. 219.
    @book {MR, MRKEY = {1937957},
      AUTHOR = {Maclachlan, Colin and Reid, Alan W.},
      TITLE = {The Arithmetic of Hyperbolic 3-Manifolds},
      SERIES = {Grad. Texts Math.},
      VOLUME = {219},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2003},
      PAGES = {xiv+463},
      ISBN = {0-387-98386-4},
      MRCLASS = {57M50 (11R52)},
      MRNUMBER = {2004i:57021},
      MRREVIEWER = {Kerry N. Jones},
      ZBLNUMBER = {1025.57001},
      }
  • [Milnor] J. Milnor, Morse Theory, Princeton, N.J.: Princeton Univ. Press, 1963, vol. 51.
    @book {Milnor, MRKEY = {0163331},
      AUTHOR = {Milnor, J.},
      TITLE = {Morse Theory},
      SERIES = {Ann. of Math. Studies},
      VOLUME={51},
      PUBLISHER = {Princeton Univ. Press},
      NOTE={based on lecture notes by M. Spivak and R. Wells},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1963},
      PAGES = {vi+153},
      MRCLASS = {57.50 (53.72)},
      MRNUMBER = {29 \#634},
      MRREVIEWER = {H. I. Levine},
      ZBLNUMBER = {0108.10401},
      }
  • [MP] Go to document T. W. Müller and J. Puchta, "Character theory of symmetric groups and subgroup growth of surface groups," J. London Math. Soc., vol. 66, iss. 3, pp. 623-640, 2002.
    @article {MP, MRKEY = {1934296},
      AUTHOR = {M{ü}ller, Thomas W. and Puchta, Jan-Christoph},
      TITLE = {Character theory of symmetric groups and subgroup growth of surface groups},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {66},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {623--640},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {20E07 (20C30 20F34 57N05)},
      MRNUMBER = {2003k:20032},
      MRREVIEWER = {Daniel Segal},
      DOI = {10.1112/S0024610702003599},
      ZBLNUMBER={1059.20021},
      }
  • [Raghunathan] M. S. Raghunathan, Discrete Subgroups of Lie Groups, New York: Springer-Verlag, 1972, vol. 68.
    @book {Raghunathan, MRKEY = {0507234},
      AUTHOR = {Raghunathan, M. S.},
      TITLE = {Discrete Subgroups of {L}ie Groups},
      SERIES= {Ergeb. Math. Grenzgeb.},
      VOLUME={68},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {ix+227},
      MRCLASS = {22E40},
      MRNUMBER = {58 \#22394a},
      MRREVIEWER = {J. S. Joel},
      ZBLNUMBER = {0254.22005},
      }
  • [Sa] B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Second ed., New York: Springer-Verlag, 2001, vol. 203.
    @book {Sa, MRKEY = {1824028},
      AUTHOR = {Sagan, Bruce E.},
      TITLE = {The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions},
      SERIES = {Grad. Texts Math.},
      VOLUME = {203},
      EDITION = {Second},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2001},
      PAGES = {xvi+238},
      ISBN = {0-387-95067-2},
      MRCLASS = {05E10 (05E05 20C30)},
      MRNUMBER = {2001m:05261},
      ZBLNUMBER = {0964.05070},
      }
  • [Th] W. P. Thurston, Three-Dimensional Geometry and Topology. Vol. 1, Princeton, NJ: Princeton Univ. Press, 1997, vol. 35.
    @book {Th, MRKEY = {1435975},
      AUTHOR = {Thurston, William P.},
      TITLE = {Three-Dimensional Geometry and Topology. {V}ol. 1},
      SERIES = {Princeton Math. Series},
      VOLUME = {35},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1997},
      PAGES = {x+311},
      ISBN = {0-691-08304-5},
      MRCLASS = {57M50 (53A35 57M25 57M60 57N10)},
      MRNUMBER = {97m:57016},
      MRREVIEWER = {Athanase Papadopoulos},
      ZBLNUMBER = {0873.57001},
      }
  • [Wang] H. C. Wang, "Topics on totally discontinuous groups," in Symmetric Spaces, New York: Dekker, 1972, vol. 8, pp. 459-487.
    @incollection {Wang, MRKEY = {0414787},
      AUTHOR = {Wang, Hsien Chung},
      TITLE = {Topics on totally discontinuous groups},
      BOOKTITLE = {Symmetric Spaces},
      VENUE={{S}hort {C}ourses, {W}ashington {U}niv., {S}t. {L}ouis, {M}o., 1969--1970},
      PAGES = {459--487},
      SERIES={Pure and Appl. Math.},
      VOLUME={8},
      PUBLISHER = {Dekker},
      ADDRESS = {New York},
      YEAR = {1972},
      MRCLASS = {22E40},
      MRNUMBER = {54 \#2879},
      MRREVIEWER = {A. L. Oniscik},
      ZBLNUMBER = {0232.22018},
      }

Authors

Mikhail Belolipetsky

Department of Mathematical Sciences
Durham University
South Road
Durham, DH1 3LE
United Kingdom
and
Sobolev Institute of Mathematics
Koptyuga 4
630090 Novosibirsk
Russia

Tsachik Gelander

Einstein Institute of Mathematics
The Hebrew University of Jerusalem
91904 Jerusalem
Israel

Alexander Lubotzky

The Hebrew University of Jerusalem
Einstein Institute of Mathematics
91904 Jerusalem
Israel

Aner Shalev

The Hebrew University of Jerusalem
Einstein Institute of Mathematics
91904 Jerusalem
Israel