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2016 Geometric generators for braid-like groups
Daniel Allcock, Tathagata Basak
Geom. Topol. 20(2): 747-778 (2016). DOI: 10.2140/gt.2016.20.747


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 n, or complex hyperbolic space n, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group PΓ. 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 PΓ 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 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 (M × M) : 2. We also suggest some other applications of our machinery.


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Daniel Allcock. Tathagata Basak. "Geometric generators for braid-like groups." Geom. Topol. 20 (2) 747 - 778, 2016.


Received: 10 March 2014; Revised: 21 April 2015; Accepted: 9 June 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1372.57004
MathSciNet: MR3493096
Digital Object Identifier: 10.2140/gt.2016.20.747

Primary: 57M05
Secondary: 20F36 , 32S22 , 52C35

Keywords: Artin groups , braid groups , Complex hyperbolic geometry , fundamental groups , infinite hyperplane arrangement , Leech lattice , Monster , presentations

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.20 • No. 2 • 2016
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