Image of the Burau representation at $d$-th roots of unity

Abstract

We show that the image of the braid group under the monodromy action on the homology of a cyclic covering of degree $d$ of the projective line is an arithmetic group provided the number of branch points is sufficiently large compared to the degree $d$. This is deduced by proving the arithmeticity of the image of the braid group on $n+1$ letters under the Burau representation evaluated at $d$-th roots of unity when $n\geq 2d$.

Note: To view the article, click on the URL link for the DOI number.

  • [Abd] Go to document M. N. Abdulrahim, "Complex specializations of the reduced Gassner representation of the pure braid group," Proc. Amer. Math. Soc., vol. 125, iss. 6, pp. 1617-1624, 1997.
    @article {Abd, MRKEY = {1422839},
      AUTHOR = {Abdulrahim, Mohammad N.},
      TITLE = {Complex specializations of the reduced {G}assner representation of the pure braid group},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {125},
      YEAR = {1997},
      NUMBER = {6},
      PAGES = {1617--1624},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {20F36 (20C15)},
      MRNUMBER = {1422839},
      MRREVIEWER = {E. Formanek},
      ZBLNUMBER = {0872.20035},
      DOI = {10.1090/S0002-9939-97-04081-1},
      }
  • [Abd2] Go to document M. N. Abdulrahim, "A faithfulness criterion for the Gassner representation of the pure braid group," Proc. Amer. Math. Soc., vol. 125, iss. 5, pp. 1249-1257, 1997.
    @article {Abd2, MRKEY = {1389500},
      AUTHOR = {Abdulrahim, Mohammad N.},
      TITLE = {A faithfulness criterion for the {G}assner representation of the pure braid group},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {125},
      YEAR = {1997},
      NUMBER = {5},
      PAGES = {1249--1257},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {20F36 (20C15)},
      MRNUMBER = {1389500},
      MRREVIEWER = {Vitaly A. Roman{\cprime}kov},
      ZBLNUMBER = {0868.20034},
      DOI = {10.1090/S0002-9939-97-03827-6},
      }
  • [A’C] N. A’Campo, "Tresses, monodromie et groupes symplectique," Comment. Math. Helv., vol. 54, pp. 318-327, 1987.
    @article{A'C,
      author={A'Campo, N.},
      TITLE={Tresses, monodromie et groupes symplectique},
      JOURNAL={Comment. Math. Helv.},
      VOLUME={54},
      YEAR={1987},
      PAGES={318--327},
      ZBLNUMBER = {0441.32004}
    }
  • [Al-Car-Tol] Go to document D. Allcock, J. A. Carlson, and D. Toledo, "The complex hyperbolic geometry of the moduli space of cubic surfaces," J. Algebraic Geom., vol. 11, iss. 4, pp. 659-724, 2002.
    @article {Al-Car-Tol, MRKEY = {1910264},
      AUTHOR = {Allcock, Daniel and Carlson, James A. and Toledo, Domingo},
      TITLE = {The complex hyperbolic geometry of the moduli space of cubic surfaces},
      JOURNAL = {J. Algebraic Geom.},
      FJOURNAL = {Journal of Algebraic Geometry},
      VOLUME = {11},
      YEAR = {2002},
      NUMBER = {4},
      PAGES = {659--724},
      ISSN = {1056-3911},
      MRCLASS = {32G20 (11F23 14J10 14J15 14K30 32M15 32Q45)},
      MRNUMBER = {1910264},
      MRREVIEWER = {I. Dolgachev},
      ZBLNUMBER = {1080.14532},
      DOI = {10.1090/S1056-3911-02-00314-4},
      }
  • [Al-Hal] Go to document D. Allcock and C. Hall, "Monodromy groups of Hurwitz-type problems," Adv. Math., vol. 225, iss. 1, pp. 69-80, 2010.
    @article {Al-Hal, MRKEY = {2669349},
      AUTHOR = {Allcock, Daniel and Hall, Chris},
      TITLE = {Monodromy groups of {H}urwitz-type problems},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {225},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {69--80},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {14H30 (14D05 57M10 57M12)},
      MRNUMBER = {2669349},
      MRREVIEWER = {Michael Friedman},
      ZBLNUMBER = {1080.14532},
      DOI = {10.1016/j.aim.2010.02.013},
      }
  • [Ba-Mi-Se] Go to document H. Bass, J. Milnor, and J. -P. Serre, "Solution of the congruence subgroup problem for ${ SL}_{n}\,(n\geq 3)$ and ${ Sp}_{2n}\,(n\geq 2)$," Inst. Hautes Études Sci. Publ. Math., vol. 33, pp. 59-137, 1967.
    @article {Ba-Mi-Se, MRKEY = {0244257},
      AUTHOR = {Bass, H. and Milnor, J. and Serre, J.-P.},
      TITLE = {Solution of the congruence subgroup problem for {${\rm SL}\sb{n}\,(n\geq 3)$} and {${\rm Sp}\sb{2n}\,(n\geq 2)$}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {33},
      YEAR = {1967},
      PAGES = {59--137},
      ISSN = {0073-8301},
      MRCLASS = {14.50 (20.00)},
      MRNUMBER = {0244257},
      MRREVIEWER = {R. Steinberg},
      ZBLNUMBER = {0174.05203},
      URL = {http://www.numdam.org/item?id=PMIHES_1967__33__59_0},
      }
  • [Beu-Hec] Go to document F. Beukers and G. Heckman, "Monodromy for the hypergeometric function $_nF_{n-1}$," Invent. Math., vol. 95, iss. 2, pp. 325-354, 1989.
    @article {Beu-Hec, MRKEY = {0974906},
      AUTHOR = {Beukers, F. and Heckman, G.},
      TITLE = {Monodromy for the hypergeometric function {$\sb nF\sb {n-1}$}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {95},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {325--354},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F99 (12H05 20F38 20G20 33A35)},
      MRNUMBER = {0974906},
      MRREVIEWER = {A. L. Onishchik},
      ZBLNUMBER = {0663.30044},
      DOI = {10.1007/BF01393900},
      }
  • [Bir] J. S. Birman, Braids, Links, and Mapping Class Groups, Princeton, N.J.: Princeton Univ. Press, 1974, vol. 82.
    @book {Bir, MRKEY = {0375281},
      AUTHOR = {Birman, Joan S.},
      TITLE = {Braids, Links, and Mapping Class Groups},
      NOTE = {Ann. of Math. Stud.},
      VOLUME={82},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1974},
      PAGES = {ix+228},
      MRCLASS = {55A25},
      MRNUMBER = {0375281},
      MRREVIEWER = {Wilbur Whitten},
      ZBLNUMBER = {0305.57013},
      }
  • [Bor-Ti] Go to document A. Borel and J. Tits, "Groupes réductifs," Inst. Hautes Études Sci. Publ. Math., vol. 27, pp. 55-150, 1965.
    @article {Bor-Ti, MRKEY = {0207712},
      AUTHOR = {Borel, Armand and Tits, Jacques},
      TITLE = {Groupes réductifs},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {27},
      YEAR = {1965},
      PAGES = {55--150},
      ISSN = {0073-8301},
      MRCLASS = {14.50 (20.75)},
      MRNUMBER = {0207712},
      MRREVIEWER = {F. D. Veldkamp},
      ZBLNUMBER = {0145.17402},
      DOI = {10.1007/BF02684375},
      }
  • [Del-Mos] Go to document P. Deligne and G. D. Mostow, "Monodromy of hypergeometric functions and nonlattice integral monodromy," Inst. Hautes Études Sci. Publ. Math., vol. 63, pp. 5-89, 1986.
    @article {Del-Mos, MRKEY = {0849651},
      AUTHOR = {Deligne, P. and Mostow, G. D.},
      TITLE = {Monodromy of hypergeometric functions and nonlattice integral monodromy},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {63},
      YEAR = {1986},
      PAGES = {5--89},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {22E40 (32G20 33A30)},
      MRNUMBER = {0849651},
      MRREVIEWER = {Shigeaki Tsuyumine},
      ZBLNUMBER = {0615.22008},
      DOI = {10.1007/BF02831622},
      }
  • [Gr-Sch] P. A. Griffiths and W. Schmid, "Recent developments in Hodge theory: a discussion of techniques and results," in Discrete Subgroups of Lie Groups and Applications to Moduli, Bombay: Oxford Univ. Press, 1975, pp. 31-127.
    @incollection{Gr-Sch,
      author={Griffiths, P. A. and Schmid, W.},
      TITLE={Recent developments in \hbox{H}odge theory: a discussion of techniques and results},
      BOOKTITLE={Discrete Subgroups of Lie Groups and Applications to Moduli},
      VENUE={Intern. Colloq., Bombay, 1973},
      YEAR={1975},
      PAGES={31--127},
      PUBLISHER={Oxford Univ. Press},
      ADDRESS={Bombay},
      MRNUMBER = {0419850},
      ZBLNUMBER = {0355.14003},
      }
  • [Gr-Lu] Go to document F. Grunewald and A. Lubotzky, "Linear representations of the automorphism group of a free group," Geom. Funct. Anal., vol. 18, iss. 5, pp. 1564-1608, 2009.
    @article {Gr-Lu, MRKEY = {2481737},
      AUTHOR = {Grunewald, Fritz and Lubotzky, Alexander},
      TITLE = {Linear representations of the automorphism group of a free group},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {18},
      YEAR = {2009},
      NUMBER = {5},
      PAGES = {1564--1608},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {20F28 (20E05 20G05)},
      MRNUMBER = {2481737},
      MRREVIEWER = {Valerij G. Bardakov},
      ZBLNUMBER = {1175.20028},
      DOI = {10.1007/s00039-009-0702-2},
      }
  • [Long] Go to document D. D. Long, "On the linear representation of braid groups," Trans. Amer. Math. Soc., vol. 311, iss. 2, pp. 535-560, 1989.
    @article {Long, MRKEY = {943606},
      AUTHOR = {Long, D. D.},
      TITLE = {On the linear representation of braid groups},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {311},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {535--560},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {20F36 (32B30 57M25)},
      MRNUMBER = {943606},
      MRREVIEWER = {J. S. Birman},
      ZBLNUMBER = {0714.20019},
      DOI = {10.2307/2001141},
      }
  • [Looi] Go to document E. Looijenga, "Uniformization by Lauricella functions—an overview of the theory of Deligne-Mostow," in Arithmetic and Geometry Around Hypergeometric Functions, Basel: Birkhäuser, 2007, vol. 260, pp. 207-244.
    @incollection {Looi, MRKEY = {2306154},
      AUTHOR = {Looijenga, Eduard},
      TITLE = {Uniformization by {L}auricella functions---an overview of the theory of {D}eligne-{M}ostow},
      BOOKTITLE = {Arithmetic and Geometry Around Hypergeometric Functions},
      SERIES = {Progr. Math.},
      VOLUME = {260},
      PAGES = {207--244},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2007},
      MRCLASS = {33C65 (32G20 33-02)},
      MRNUMBER = {2306154},
      MRREVIEWER = {Martin Deraux},
      ZBLNUMBER = {1120.33013},
      DOI = {10.1007/978-3-7643-8284-1_8},
      }
  • [Looi2] Go to document E. Looijenga, "Prym representations of mapping class groups," Geom. Dedicata, vol. 64, iss. 1, pp. 69-83, 1997.
    @article {Looi2, MRKEY = {1432535},
      AUTHOR = {Looijenga, Eduard},
      TITLE = {Prym representations of mapping class groups},
      JOURNAL = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {64},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {69--83},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {57M10 (14H10 20F38 57N05)},
      MRNUMBER = {1432535},
      MRREVIEWER = {Ronnie Lee},
      ZBLNUMBER = {0872.57018},
      DOI = {10.1023/A:1004909416648},
      }
  • [Mar] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, New York: Springer-Verlag, 1991, vol. 17.
    @book {Mar, MRKEY = {1090825},
      AUTHOR = {Margulis, G. A.},
      TITLE = {Discrete Subgroups of Semisimple {L}ie Groups},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {17},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      PAGES = {x+388},
      ISBN = {3-540-12179-X},
      MRCLASS = {22E40 (20Hxx 22-02 22D40)},
      MRNUMBER = {1090825},
      MRREVIEWER = {Gopal Prasad},
      ZBLNUMBER = {0732.22008},
      }
  • [Mc] Go to document C. T. McMullen, "Braid groups and Hodge theory," Math. Ann., vol. 355, iss. 3, pp. 893-946, 2013.
    @article {Mc, MRKEY = {3020148},
      AUTHOR = {McMullen, Curtis T.},
      TITLE = {Braid groups and {H}odge theory},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {355},
      YEAR = {2013},
      NUMBER = {3},
      PAGES = {893--946},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {20F36 (14C30)},
      MRNUMBER = {3020148},
      ZBLNUMBER = {06149478},
      DOI = {10.1007/s00208-012-0804-2},
      }
  • [Mos] Go to document G. D. Mostow, "Generalized Picard lattices arising from half-integral conditions," Inst. Hautes Études Sci. Publ. Math., vol. 63, pp. 91-106, 1986.
    @article {Mos, MRKEY = {0849652},
      AUTHOR = {Mostow, G. D.},
      TITLE = {Generalized {P}icard lattices arising from half-integral conditions},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {63},
      YEAR = {1986},
      PAGES = {91--106},
      ISSN = {0073-8301},
      CODEN = { PMIHA6},
      MRCLASS = {22E40 (32G20 33A30)},
      MRNUMBER = {0849652},
      MRREVIEWER = {Shigeaki Tsuyumine},
      ZBLNUMBER = {0615.22009},
      DOI = {10.1007/BF02831623},
      }
  • [Nor] M. V. Nori, "A nonarithmetic monodromy group," C. R. Acad. Sci. Paris Sér. I Math., vol. 302, iss. 2, pp. 71-72, 1986.
    @article {Nor, MRKEY = {0832040},
      AUTHOR = {Nori, Madhav V.},
      TITLE = {A nonarithmetic monodromy group},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {302},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {71--72},
      ISSN = {0249-6291},
      CODEN = {CASMEI},
      MRCLASS = {14D05 (14K15)},
      MRNUMBER = {0832040},
      MRREVIEWER = {J. S. Joel},
      ZBLNUMBER = {0602.14025},
      }
  • [Pl-Ra] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Boston, MA: Academic Press Inc., 1994, vol. 139.
    @book{Pl-Ra, MRKEY={1278263},
      AUTHOR = {Platonov, Vladimir and Rapinchuk, Andrei},
      TITLE = {Algebraic Groups and Number Theory},
      SERIES = {Pure and Applied Mathematics},
      VOLUME = {139},
      NOTE = {translated from the 1991 Russian original by Rachel Rowen},
      PUBLISHER = {Academic Press Inc.},
      ADDRESS = {Boston, MA},
      YEAR = {1994},
      PAGES = {xii+614},
      ISBN = {0-12-558180-7},
      MRCLASS = {11E57 (11-02 20Gxx)},
      MRNUMBER = {1278263},
      ZBLNUMBER = {0841.20046},
     }
  • [Ra] Go to document M. S. Raghunathan, "A note on generators for arithmetic subgroups of algebraic groups," Pacific J. Math., vol. 152, iss. 2, pp. 365-373, 1992.
    @article {Ra, MRKEY = {1141802},
      AUTHOR = {Raghunathan, M. S.},
      TITLE = {A note on generators for arithmetic subgroups of algebraic groups},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {152},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {365--373},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {22E40 (20H20)},
      MRNUMBER = {1141802},
      MRREVIEWER = {Avner Ash},
      ZBLNUMBER = {0793.20045},
      DOI = {10.2140/pjm.1992.152.365},
     }
  • [Ra2] Go to document M. S. Raghunathan, Discrete Subgroups of Lie Groups, New York: Springer-Verlag, 1972, vol. 68.
    @book {Ra2, 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 = {0507234},
      MRREVIEWER = {J. S. Joel},
      ZBLNUMBER = {0254.22005},
      DOI = {10.2140/pjm.1992.152.365},
      }
  • [Sar] Go to document P. Sarnak, Notes on thin groups, 2012.
    @misc{Sar,
      author={Sarnak, P.},
      TITLE={Notes on thin groups},
      NOTE={MSRI Hot Topics Workshop},
      YEAR={2012} , URL ={http://www.msri.org/attachments/workshops/652_Sarnak-notes.pdf},
     }
  • [Sq] Go to document C. C. Squier, "The Burau representation is unitary," Proc. Amer. Math. Soc., vol. 90, iss. 2, pp. 199-202, 1984.
    @article {Sq, MRKEY = {0727232},
      AUTHOR = {Squier, Craig C.},
      TITLE = {The {B}urau representation is unitary},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {90},
      YEAR = {1984},
      NUMBER = {2},
      PAGES = {199--202},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {20F36},
      MRNUMBER = {0727232},
      MRREVIEWER = {J. S. Birman},
      ZBLNUMBER = {0542.20022},
      DOI = {10.2307/2045338},
      }
  • [Ti] J. Tits, "Systèmes générateurs de groupes de congruence," C. R. Acad. Sci. Paris Sér. A-B, vol. 283, iss. 9, p. ai, a693-a695, 1976.
    @article {Ti, MRKEY = {0424966},
      AUTHOR = {Tits, Jacques},
      TITLE = {Systèmes générateurs de groupes de congruence},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
      VOLUME = {283},
      YEAR = {1976},
      NUMBER = {9},
      PAGES = {Ai, A693--A695},
      MRCLASS = {20H05 (20G30)},
      MRNUMBER = {0424966},
      MRREVIEWER = {J. E. Humphreys},
      ZBLNUMBER = {0381.14005},
      }
  • [Va] Go to document L. N. Vaservsteuin, "Structure of the classical arithmetic groups of rank greater than $1$," Mat. Sb., vol. 20, pp. 465-492, 1973.
    @article {Va, MRKEY = {0349864},
      AUTHOR = {Vaser{š}te{\u\i}n, L. N.},
      TITLE = {Structure of the classical arithmetic groups of rank greater than {$1$}},
      JOURNAL = {Mat. Sb.},
      VOLUME = {20},
      YEAR = {1973},
      PAGES = {465-492},
      MRCLASS = {20H05 (20H10)},
      MRNUMBER = {0349864},
      MRREVIEWER = {K. W. Roggenkamp},
      ZBLNUMBER = {0291.14016},
      DOI = {10.1070/SM1973v020n03ABEH001885},
      }
  • [Ve] Go to document T. N. Venkataramana, "On systems of generators of arithmetic subgroups of higher rank groups," Pacific J. Math., vol. 166, iss. 1, pp. 193-212, 1994.
    @article {Ve, MRKEY = {1306038},
      AUTHOR = {Venkataramana, T. N.},
      TITLE = {On systems of generators of arithmetic subgroups of higher rank groups},
      JOURNAL = {Pacific J. Math.},
      FJOURNAL = {Pacific Journal of Mathematics},
      VOLUME = {166},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {193--212},
      ISSN = {0030-8730},
      CODEN = {PJMAAI},
      MRCLASS = {22E40 (20G15)},
      MRNUMBER = {1306038},
      MRREVIEWER = {Alexander Starkov},
      ZBLNUMBER = {0822.22005},
      DOI = {10.2140/pjm.1994.166.193},
      }

Authors

T. N. Venkataramana

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai, 40005, India