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

T. N. Venkataramana

doi:https://doi.org/10.4007/annals.2014.179.3.4

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 $n\geq 2d$.

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[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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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}, }

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

Abstract

Authors