Geometry & Topology

Geometric generators for braid-like groups

Abstract

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 $ℂn$, or complex hyperbolic space $ℂℍn$, 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 $ℂℍ13$. 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.

Article information

Source
Geom. Topol., Volume 20, Number 2 (2016), 747-778.

Dates
Revised: 21 April 2015
Accepted: 9 June 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858970

Digital Object Identifier
doi:10.2140/gt.2016.20.747

Mathematical Reviews number (MathSciNet)
MR3493096

Zentralblatt MATH identifier
1372.57004

Citation

Allcock, Daniel; Basak, Tathagata. Geometric generators for braid-like groups. Geom. Topol. 20 (2016), no. 2, 747--778. doi:10.2140/gt.2016.20.747. https://projecteuclid.org/euclid.gt/1510858970

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