Congruence subgroups of braid groups

Tara E. Brendle

doi:10.5802/wbln.23

Tara E. Brendle ¹

¹ Tara E. Brendle, School of Mathematics & Statistics, University Place, University of Glasgow, G12 8SQ

Winter Braids Lecture Notes, Volume 5 (2018), Talk no. 3, 26 p.

Abstract

These notes are based on a mini-course given at CIRM in February 2018 as part of the workshop Winter Braids VIII.

Published online: 2020-06-29

Zbl: 06836109

DOI: 10.5802/wbln.23

Author's affiliations:

Tara E. Brendle ¹

¹ Tara E. Brendle, School of Mathematics & Statistics, University Place, University of Glasgow, G12 8SQ

@article{WBLN_2018__5__A3_0,
     author = {Tara E. Brendle},
     title = {Congruence subgroups of braid groups},
     journal = {Winter Braids Lecture Notes},
     note = {talk:3},
     publisher = {Winter Braids School},
     volume = {5},
     year = {2018},
     doi = {10.5802/wbln.23},
     zbl = {06836109},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/wbln.23/}
}

TY  - JOUR
AU  - Tara E. Brendle
TI  - Congruence subgroups of braid groups
JO  - Winter Braids Lecture Notes
N1  - talk:3
PY  - 2018
VL  - 5
PB  - Winter Braids School
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/wbln.23/
DO  - 10.5802/wbln.23
LA  - en
ID  - WBLN_2018__5__A3_0
ER  -

%0 Journal Article
%A Tara E. Brendle
%T Congruence subgroups of braid groups
%J Winter Braids Lecture Notes
%Z talk:3
%D 2018
%V 5
%I Winter Braids School
%U https://proceedings.centre-mersenne.org/articles/10.5802/wbln.23/
%R 10.5802/wbln.23
%G en
%F WBLN_2018__5__A3_0

Tara E. Brendle. Congruence subgroups of braid groups. Winter Braids Lecture Notes, Volume 5 (2018), Talk no. 3, 26 p. doi : 10.5802/wbln.23. https://proceedings.centre-mersenne.org/articles/10.5802/wbln.23/

References
Cited by

[1] Norbert A’Campo Tresses, monodromie et le groupe symplectique, Comment. Math. Helv., Volume 54 (1979) no. 2, pp. 318-327 | DOI | MR | Zbl

[2] Jessica Appel; Wade Bloomquist; Katie Gravel; Annie Holden On Congruence Subgroups of the Braid Group (https://people.math.gatech.edu/~dmargalit7/reu.shtml/Braided_Poster.pdf)

[3] V. I. Arnol’d A remark on the branching of hyperelliptic integrals as functions of the parameters, Funkcional. Anal. i Priložen., Volume 2 (1968) no. 3, pp. 1-3 | MR | Zbl

[4] E. Artin Theory of braids, Ann. of Math. (2), Volume 48 (1947), pp. 101-126 | DOI | MR | Zbl

[5] Emil Artin Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, Volume 4 (1925) no. 1, pp. 47-72 | DOI | MR | Zbl

[6] Joachim Assion Einige endliche Faktorgruppen der Zopfgruppen, Math. Z., Volume 163 (1978) no. 3, pp. 291-302 | DOI | MR | Zbl

[7] Stephen J. Bigelow; Ryan D. Budney The mapping class group of a genus two surface is linear, Algebr. Geom. Topol., Volume 1 (2001), pp. 699-708 | DOI | MR | Zbl

[8] Joan S. Birman On Siegel’s modular group, Math. Ann., Volume 191 (1971), pp. 59-68 | DOI | MR | Zbl

[9] Joan S. Birman Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974, ix+228 pages (Annals of Mathematics Studies, No. 82) | MR | Zbl

[10] Joan S. Birman; Hugh M. Hilden On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. (2), Volume 97 (1973), pp. 424-439 https://doi-org.prx.library.gatech.edu/10.2307/1970830 | DOI | MR | Zbl

[11] Tara Brendle; Dan Margalit; Andrew Putman Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at $t = - 1$ , Invent. Math., Volume 200 (2015) no. 1, pp. 263-310 | DOI | MR | Zbl

[12] Tara E. Brendle; Dan Margalit Corrigendum to: The level four braid group (Preprint in preparation.) | Zbl

[13] Tara E. Brendle; Dan Margalit Point pushing, homology, and the hyperelliptic involution, Michigan Math. J., Volume 62 (2013) no. 3, pp. 451-473 | DOI | MR | Zbl

[14] Tara E. Brendle; Dan Margalit The level four braid group, J. Reine Angew. Math., Volume 735 (2018), pp. 249-264 | DOI | MR | Zbl

[15] F. R. Cohen; J. Wu On braid groups and homotopy groups, Groups, homotopy and configuration spaces (Geom. Topol. Monogr.), Volume 13, Geom. Topol. Publ., Coventry, 2008, pp. 169-193 | DOI | MR | Zbl

[16] Max Dehn Papers on group theory and topology, Springer-Verlag, New York, 1987, viii+396 pages (Translated from the German and with introductions and an appendix by John Stillwell, With an appendix by Otto Schreier) | DOI | MR | Zbl

[17] Steven Diaz; Ron Donagi; David Harbater Every curve is a Hurwitz space, Duke Math. J., Volume 59 (1989) no. 3, pp. 737-746 https://doi-org.ezproxy.lib.gla.ac.uk/10.1215/S0012-7094-89-05933-4 | DOI | MR | Zbl

[18] Benson Farb; Dan Margalit A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012, xiv+472 pages | MR | Zbl

[19] Neil Fullarton; Andrew Putman The high-dimensional cohomology of the moduli space of curves with level structures, J. Eur. Math. Soc. (to appear) | Zbl

[20] Neil J. Fullarton A generating set for the palindromic Torelli group, Algebr. Geom. Topol., Volume 15 (2015) no. 6, pp. 3535-3567 | DOI | MR | Zbl

[21] Jean-Marc Gambaudo; Étienne Ghys Braids and signatures, Bull. Soc. Math. France, Volume 133 (2005) no. 4, pp. 541-579 | DOI | Numdam | MR | Zbl

[22] Edna K. Grossman On the residual finiteness of certain mapping class groups, J. London Math. Soc. (2), Volume 9 (1974/75), pp. 160-164 | DOI | MR | Zbl

[23] Richard Hain Finiteness and Torelli spaces, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.), Volume 74, Amer. Math. Soc., Providence, RI, 2006, pp. 57-70 | DOI | MR | Zbl

[24] John L. Harer The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., Volume 84 (1986) no. 1, pp. 157-176 | DOI | MR | Zbl

[25] Stephen P. Humphries Generators for the mapping class group, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) (Lecture Notes in Math.), Volume 722, Springer, Berlin, 1979, pp. 44-47 | DOI | MR | Zbl

[26] N. V. Ivanov Residual finiteness of modular Teichmüller groups, Sibirsk. Mat. Zh., Volume 32 (1991) no. 1, p. 182-185, 222 | DOI | MR

[27] Nikolai V. Ivanov Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115, American Mathematical Society, Providence, RI, 1992, xii+127 pages (Translated from the Russian by E. J. F. Primrose and revised by the author) | MR

[28] Nikolai V. Ivanov Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices, Volume 1997 (1997) no. 14, pp. 651-666 | DOI | MR | Zbl

[29] Dennis Johnson An abelian quotient of the mapping class group $ℐ_{g}$ , Math. Ann., Volume 249 (1980) no. 3, pp. 225-242 https://doi-org.prx.library.gatech.edu/10.1007/BF01363897 | DOI | MR | Zbl

[30] Dennis Johnson The structure of the Torelli group. I. A finite set of generators for $ℐ$ , Ann. of Math. (2), Volume 118 (1983) no. 3, pp. 423-442 https://doi-org.prx.library.gatech.edu/10.2307/2006977 | DOI | MR | Zbl

[31] Dennis Johnson The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology, Volume 24 (1985) no. 2, pp. 113-126 https://doi-org.prx.library.gatech.edu/10.1016/0040-9383(85)90049-7 | DOI | MR | Zbl

[32] JSE The image of the point-pushing group in the hyperelliptic representation of the braid group (MathOverflow, https://mathoverflow.net/questions/105048/the-image-of-the-point-pushing-group-in-the-hyperelliptic-representation-of-the) | Zbl

[33] Kevin Kordek; Dan Margalit Representation stability in the level 4 braid group (arXiv:1903.03119)

[34] Mustafa Korkmaz Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl., Volume 95 (1999) no. 2, pp. 85-111 | DOI | MR | Zbl

[35] Catherine Labruère; Luis Paris Presentations for the punctured mapping class groups in terms of Artin groups, Algebr. Geom. Topol., Volume 1 (2001), pp. 73-114 https://doi-org.ezproxy.lib.gla.ac.uk/10.2140/agt.2001.1.73 | DOI | MR | Zbl

[36] Feng Luo Automorphisms of the complex of curves, Topology, Volume 39 (2000) no. 2, pp. 283-298 | DOI | MR | Zbl

[37] Dan Margalit; Rebecca Winarski BIrman-Hilden theory, Celebratio Mathematica (To appear)

[38] D. B. McReynolds The congruence subgroup problem for pure braid groups: Thurston’s proof, New York J. Math., Volume 18 (2012), pp. 925-942 http://nyjm.albany.edu:8000/j/2012/18_925.html | MR | Zbl

[39] J. Mennicke Zur Theorie der Siegelschen Modulgruppe, Math. Ann., Volume 159 (1965), pp. 115-129 | DOI | MR | Zbl

[40] Geoffrey Mess The Torelli groups for genus $2$ and $3$ surfaces, Topology, Volume 31 (1992) no. 4, pp. 775-790 https://doi-org.prx.library.gatech.edu/10.1016/0040-9383(92)90008-6 | DOI | MR | Zbl

[41] Takayuki Morifuji On Meyer’s function of hyperelliptic mapping class groups, J. Math. Soc. Japan, Volume 55 (2003) no. 1, pp. 117-129 | DOI | MR | Zbl

[42] Morris Newman Integral matrices, Academic Press, New York-London, 1972, xvii+224 pages (Pure and Applied Mathematics, Vol. 45) | MR | Zbl

[43] Jerome Powell Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc., Volume 68 (1978) no. 3, pp. 347-350 | DOI | MR | Zbl

[44] Andrew Putman Lectures on the Torelli group (https://www3.nd.edu/ andyp/teaching/2014SpringMath541/) | Zbl

[45] Andrew Putman The Torelli group and congruence subgroups of the mapping class group (https://www3.nd.edu/ andyp/notes/) | MR | Zbl

[46] Andrew Putman Cutting and pasting in the Torelli group, Geom. Topol., Volume 11 (2007), pp. 829-865 | DOI | MR | Zbl

[47] Andrew Putman An infinite presentation of the Torelli group, Geom. Funct. Anal., Volume 19 (2009) no. 2, pp. 591-643 | DOI | MR | Zbl

[48] Andrew Putman Obtaining presentations from group actions without making choices, Algebr. Geom. Topol., Volume 11 (2011) no. 3, pp. 1737-1766 | DOI | MR | Zbl

[49] M. S. Raghunathan The congruence subgroup problem, Proc. Indian Acad. Sci. Math. Sci., Volume 114 (2004) no. 4, pp. 299-308 | DOI | MR | Zbl

[50] Dale Rolfsen Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976, ix+439 pages (Mathematics Lecture Series, No. 7) | MR | Zbl

[51] H. L. Royden Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) (Ann. of Math. Studies), Volume 66 (1971), pp. 369-383 | DOI | MR | Zbl

[52] Nancy Scherich Classification of the real discrete specialisations of the Burau representation of B3, Math. Proc. Cambridge Philos. Soc., Volume 168 (2020) no. 2, pp. 295-304 https://doi-org.ezproxy.lib.gla.ac.uk/10.1017/s0305004118000683 | DOI | MR | Zbl

[53] Craig C. Squier The Burau representation is unitary, Proc. Amer. Math. Soc., Volume 90 (1984) no. 2, pp. 199-202 | DOI | MR | Zbl

[54] Charalampos Stylianakis Congruence subgroups of braid groups, Internat. J. Algebra Comput., Volume 28 (2018) no. 2, pp. 345-364 | DOI | MR | Zbl

[55] J. G. Sunday Presentations of the groups $SL (2, m)$ and $PSL (2, m)$ , Canadian J. Math., Volume 24 (1972), pp. 1129-1131 | DOI | MR | Zbl

[56] Vladimir Turaev Faithful linear representations of the braid groups, Astérisque (2002) no. 276, pp. 389-409 (Séminaire Bourbaki, Vol. 1999/2000) | Numdam | MR | Zbl

[57] Bronislaw Wajnryb A braidlike presentation of $Sp (n, p)$ , Israel J. Math., Volume 76 (1991) no. 3, pp. 265-288 | DOI | MR | Zbl

[58] J.-K. Yu Toward a proof of the Cohen-Lenstra conjecture in the function field case (1996) (Preprint.)

Cited by Sources: