Geometry & Topology
- Geom. Topol.
- Volume 20, Number 2 (2016), 747-778.
Geometric generators for braid-like groups
We study the problem of finding generators for the fundamental group of a space of the following sort: one removes a family of complex hyperplanes from , or complex hyperbolic space , or the Hermitian symmetric space for , and then takes the quotient by a discrete group . 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 contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for in a particular intricate example in . The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group , that gives geometric meaning to the generators and relations in the Conway–Simons presentation of . We also suggest some other applications of our machinery.
Geom. Topol., Volume 20, Number 2 (2016), 747-778.
Received: 10 March 2014
Revised: 21 April 2015
Accepted: 9 June 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20F36: Braid groups; Artin groups 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 32S22: Relations with arrangements of hyperplanes [See also 52C35]
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