Geometric generators for braid-like groups

Allcock, Daniel¹ ; Basak, Tathagata²

¹ Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-1082, USA
² Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Geometry & topology, Tome 20 (2016) no. 2, pp. 747-778.

Résumé

We study the problem of finding generators for the fundamental group $G$ of a space of the following sort: one removes a family of complex hyperplanes from ${ℂ}^n$, or complex hyperbolic space $ℂ{ℍ}^n$, or the Hermitian symmetric space for $O\left(2,n\right)$, and then takes the quotient by a discrete group $P\Gamma$. The classical example is the braid group, but there are many similar “braid-like” groups that arise in topology and algebraic geometry. Our main result is that if $P\Gamma$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then $G$ is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for $G$ in a particular intricate example in $ℂ{ℍ}^{13}$. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group $M$, that gives geometric meaning to the generators and relations in the Conway–Simons presentation of $\left(M\times M\right):2$. We also suggest some other applications of our machinery.

Détail

DOI : 10.2140/gt.2016.20.747

Classification : 57M05, 20F36, 52C35, 32S22
Keywords: fundamental groups, infinite hyperplane arrangement, complex hyperbolic geometry, braid groups, Artin groups, Leech lattice, presentations, Monster

Bibliographie

[1] D Allcock, The Leech lattice and complex hyperbolic reflections, Invent. Math. 140 (2000) 283

[2] D Allcock, New complex- and quaternion-hyperbolic reflection groups, Duke Math. J. 103 (2000) 303

[3] D Allcock, The period lattice for Enriques surfaces, Math. Ann. 317 (2000) 483

[4] D Allcock, A monstrous proposal, from: "Groups and symmetries" (editors J Harnad, P Winternitz), CRM Proc. Lecture Notes 47, Amer. Math. Soc. (2009) 17

[5] D Allcock, On the Y 555 complex reflection group, J. Algebra 322 (2009) 1454

[6] D Allcock, T Basak, Generators for a complex hyperbolic braid group

[7] D Allcock, J A Carlson, D Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geom. 11 (2002) 659

[8] D Allcock, J A Carlson, D Toledo, The moduli space of cubic threefolds as a ball quotient, Amer. Math. Soc. (2011)

[9] T Basak, The complex Lorentzian Leech lattice and the bimonster, J. Algebra 309 (2007) 32

[10] T Basak, The complex Lorentzian Leech lattice and the bimonster, II, Trans. Amer. Math. Soc. 368 (2016) 4171

[11] D Bessis, Finite complex reflection arrangements are K(π,1), Ann. of Math. 181 (2015) 809

[12] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999)

[13] E Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971) 57

[14] J H Conway, A group of order 8,315,553,613,086,720,000, Bull. London Math. Soc. 1 (1969) 79

[15] J H Conway, R A Parker, N J A Sloane, The covering radius of the Leech lattice, Proc. Roy. Soc. London Ser. A 380 (1982) 261

[16] J H Conway, C S Simons, 26 implies the Bimonster, J. Algebra 235 (2001) 805

[17] H S M Coxeter, Finite groups generated by unitary reflections, Abh. Math. Sem. Univ. Hamburg 31 (1967) 125

[18] I V Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci. 81 (1996) 2599

[19] R Fox, L Neuwirth, The braid groups, Math. Scand. 10 (1962) 119

[20] S Gallot, D Hulin, J Lafontaine, Riemannian geometry, Springer (1987)

[21] W M Goldman, Complex hyperbolic geometry, Clarendon Press (1999)

[22] M Goresky, R Macpherson, Stratified Morse theory, 14, Springer (1988)

[23] G Heckman, The Allcock ball quotient, preprint (2013)

[24] G Heckman, E Looijenga, The moduli space of rational elliptic surfaces, from: "Algebraic geometry 2000, Azumino" (editors S Usui, M Green, L Illusie, K Kato, E Looijenga, S Mukai, S Saito), Adv. Stud. Pure Math. 36, Math. Soc. Japan (2002) 185

[25] G Heckman, S Rieken, Two Lorentzian lattices, preprint (2014)

[26] M Kapovich, Hyperbolic manifolds and discrete groups, 183, Birkhäuser (2001)

[27] S Kondō, A complex hyperbolic structure for the moduli space of curves of genus three, J. Reine Angew. Math. 525 (2000) 219

[28] S Kondō, The moduli space of curves of genus 4 and Deligne–Mostow’s complex reflection groups, from: "Algebraic geometry 2000, Azumino" (editors S Usui, M Green, L Illusie, K Kato, E Looijenga, S Mukai, S Saito), Adv. Stud. Pure Math. 36, Math. Soc. Japan (2002) 383

[29] E Looijenga, On the semi-universal deformation of a simple-elliptic hypersurface singularity : Unimodularity, Topology 16 (1977) 257

[30] E Looijenga, The smoothing components of a triangle singularity, II, Math. Ann. 269 (1984) 357

[31] E Looijenga, Artin groups and the fundamental groups of some moduli spaces, J. Topol. 1 (2008) 187

[32] E Looijenga, The period map for cubic fourfolds, Invent. Math. 177 (2009) 213

[33] E Looijenga, R Swierstra, The period map for cubic threefolds, Compos. Math. 143 (2007) 1037

[34] Y Namikawa, Periods of Enriques surfaces, Math. Ann. 270 (1985) 201

[35] V V Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces, Trudy Moskov. Mat. Obshch. 38 (1979) 75

[36] J G Ratcliffe, Foundations of hyperbolic manifolds, 149, Springer (1994)

[37] G C Shephard, J A Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954) 274

[38] R A Wilson, The complex Leech lattice and maximal subgroups of the Suzuki group, J. Algebra 84 (1983) 151