Geometry & Topology

Geometric generators for braid-like groups

Daniel Allcock and Tathagata Basak

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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.

Article information

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

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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]

fundamental groups infinite hyperplane arrangement complex hyperbolic geometry braid groups Artin groups Leech lattice presentations Monster


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.

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